Three parts to this answer- Why I spent three weeks hacking a system to do exactly this, why FATE is a good base system for this, a detail on the hacks I used, and examples. A TLDR will be at the end. This is mostly spoiler free, with the one big spoiler clearly marked, and everything else ambiguous enough I don't think it should ruin anybody.
The end of Worm coincided very close with the beginning of a 5 week winter break, and the gang back at home asking me to run a game. I did a whipround couple of questions of what people wanted. Requests were strongly in favour of a superhero game, with other elements being light-heartedness, Pacific Rim influences, and a system they hadn't used in a year or so. (So no D&D 3.5 or Werewolf: The Apocalypse.) I knew as soon as they said they wanted superheroes that I wanted to make this as much like Worm as they would stand for. (none of my players read it, which gave me a lot of freedom to mix and match what I wanted from it.) I knew several of my players did that experiences in high school much like the first few chapters of Worm, so trigger events were getting reworked. I knew my group tended to like a strong equal playing field, disliked long lists or tables, liked having a tangible effect on the game world, were primarily simulationist with a smattering of narrativism and one lone diehard gamist, and liked having a tightnit group. The desire for light-heartedness is what made the biggest changes from the wormverse.
I needed a system that could deal quickly and easily with wild, crazy powers without a huge bunch of tables, could function 'realistically' while still allowing for character driven play. I had to have enough of a game in there to satisfy the gamist, without letting him get in anyone else's way. And since I knew nothing perfectly fit Worm, it had to be conducive to hacking.
The above only matters to you in that it means that as I was hacking FATE, I was trying to achieve those goals. I'll point out places where my hacks make no sense for Worm, but do achieve things my group needed.
Why FATE works for this
So, the things you specified as being required for Worm are moral complexity, large variety of unique powers, the category system, and creative use of powers. These are not exactly the things I think of as central to Worm, but fortunately FATE works for both your list and mine, so I'm going to go over both.
Moral complexity will always be at least as much of a product of your gaming group as it will be a product of the system- there are systems that hamper you in reaching this (and which systems these are can be kind of flamebait) and there are also systems that hope for it (Sorcerer comes to mind, as does Dogs in the Vineyard, both of which I'm considering writing up as separate answers.) In FATE, these shades of grey will likely come from Aspects (which I'll be talking about later again.)
An aspect is a short, catchy sentence or descriptor of a character. They have both narrative and mechanical impact, and separating the two can be tricky. They can be compelled, meaning that they are causing a problem for the player, and they can be invoked, meaning they are being useful to the player. Taylor could have an aspect like "Secretly a superhero" and another aspect of "The Newest Undersider" and any DM worth their salt would compel both of these aspects liberally at different times, causing the see-saw between Taylor's interests in the first chunk of the story. Contradictory aspects are wonderful for creating a complex character, but I will say that the system doesn't force you into that. As a DM, if you want moral ambiguity, you should encourage your players to take aspects both good and bad during character creation.
The other cool thing about aspects is that they are always true. This means that, aside from being good bits of characterization, they double as the core of the powers system. Someone with the aspect "Metal Body" is going to be impossible to hurt using normal weapons, and the system is just fine with that, while someone else could have the aspect "Macrohydrokinetic" which will also function just fine. (A bit of advice- Be sure that both the players and the DM are on the same page about what a power aspect does.) This means that literally any power your players and you can imagine can be used in the game. I prefer this far more than using a list of powers as building blocks- Any list of powers is going to subtly encourage players towards something that fits with the list.
These aspects can be tagged or just provide justification for anything that the DM and the players agree it does. While this does mean that a bit more negotiation might occur than if you were using a hard and fast ruleset, it also means you have far more room to be creative with it. I can't think of any RPG that would have hard and fast rules for say, jamming a robot with bug guts to stop it working, but such a tactic could be very effective, so I think you are going to have to go for something that eschews the hard and fast.
Honestly, I don't agree greatly with the classification system you use in your question. It's an in-universe system made by people who did not know everything about powers, which various characters occasionally comment as being sometimes unsuited for what it's being used for. (In the chapter where we learn about the classification system in fact, we quickly hear that it may not be good at classifying the character who lists off the rhyme.) However- "Equivalent Exchange Teleporter" is a good aspect for a mover, "My Mind's Labyrinth" would work for a shaker, "Metal Body" fits my favourite brute, "Giant Form" works for the only straight up breakers I remember, "Human Nervous System Hijacker" works on both levels for a genial sociopathic master, "Mad Genius Bomb Maker" does nicely for a crazy tinker, "Time-stop Touch" is a well known striker, etc. Point is, you can come up with whatever powers you want to, and classify them after if you like.
Things I like about Worm that FATE helps with include, among others, the way injuries stack up on a character. Each sizeable blow in FATE results in a Consequence- a temporary aspect like "Broken leg" or "Bruised." They can be compelled just like other aspects, resulting in situations where, say, an injured arm prevents you from being able to run from the rampaging kaiju effectively. They also aren't all physical- mental trauma also can be a consequence. What I absolutely love is, an extreme consequence (something like "Flayed alive" or "My brother's Suicide") mandates altering or replacing a regular aspect of your character sheet, and can take multiple story arcs to fade and can still leave the altered aspect behind. This fits trigger events so perfectly, I'm kind of disappointed I didn't want to use this on my players. (Explaining second trigger events would mean extra lore, but more importantly there were members of my gaming group who had actually been through some trigger-worthy events. Please get some kind of okay from your players before you use this. Gestation nearly made me lose my lunch, and I was a physically adept guy.)
For all of that though, FATE is highly non-lethal. While you can certainly die, it's fairly common for someone with no defensive powers to have a truck thrown at their head, and survive due to circumstance. Seriously, people in Worm get horribly injured and maimed but the actual number of people who bite the dust is far lower than you'd expect given that the very first high profile fight involves multi-ton attack dogs, far too many venomous spiders for anyone's comfort, a gigantic laser cannon, and someone who can onehandedly flip a truck. You even have Concessions- a metagame surrender, often you taking some serious injury then retreating from the battlefield. The default mode of FATE is that PCs don't die, they just get captured or defeated. It's considered good form to tell the players when death is on the line, and consequences will not be accepted- Something very analogous to Wormverse Kill orders. Fighting with the Wards, nobody's going to die. When the Nine come to town, everyone knows that slipping up can mean death. You can move back and forth between deadly and non-deadly combat, sometimes even against the same antagonists or in special circumstances. "Shadow Stalker's arrived- you know the rest of the wards won't kill you, but Grue, if she can kill you, she will."
I also like that, if you go through the full world creation, you wind up creating a city with areas each with a character associated with it, and the players are likely to have an aspect to do with that area. The Docks are wonderfully realized in Worm, and some of my favourite parts of the story are where The Docks(The Place) Danny Herbert (The Face) and Skitter (The Player) are at their most bound up. The PRT HQ gets represented quite strongly by the character of Armsmaster. When it came time for my players to divide up the turf they planned to guard, they all knew what part of the city they wanted control of and exactly how they wanted to shape it.
Lastly is the way distance is measured. You draw a rough map of the area, then divide it into zones based on distance and ease of movement. This seems like a small detail, but you can rescale the map however you want without changing how you're doing things. This means that a street fight about a bank robbery, a citywide battle against godzilla
and a multiverse spanning war against a god
can be handled with the same system of maps and movement, should the need arise. Heck, combat as a whole will work the same way on all of those scales. And every part of that map will wind up strewn with aspects like "On fire." After my group faced off with Jack Slash and Burnscar, they looked down and the map and just laughed at how completely wrecked the area was. I sicked Leviathan on them last night for the final session, and I didn't have to teach them a whole new mass combat system to deal with that.
(Note- this section assumes that you are convinced that FATE is the way to go for Wormverse games, and is therefore detailing what I did to make FATE fit the 'verse more closely. It assumes you've gotten your hands on FATE Core or FATE Accelerated Edition and given them at least a basic read.)
We did full city creation- First, come up with the names of places in the city and two aspects for that place. Then come up with a person that fits for that place, and two aspects for them. When making characters, look at having an aspect to do with a place or a person. Straight and simple. What it gives us is a world that all players are invested in- Taylor is tied up with her dad, who is tied up with the docks. Grue has a thing with Shadow Stalker, who is a good expression of everything that's wrong with the PRT.
For characters, start with FATE Core.
Skill list is Power, Genius, Acrobatics, Taunt, Charm, Identity, and Brawl. They function a little like approaches from Fate Accelerated, in that they cover anything close to them and are more ways of dealing with a problem that can be attacked from any skill.
Brawl handles most fisticuffs, gunfire, etc. Would be Fight in most FATE games, but I wanted to make it clear other skills could be used for combat, and it reinforced the method of fighting most common- almost nobody in Worm seems to have real martial arts training, and guns are oddly infrequent. Identity is what your alternate identity is and how well it's kept- Rachel has a very low Identity, Taylor has a medium identity, Coil has a fantastic Identity. Honestly, if I was going to do it again, I might drop Identity from the lineup. Charm is getting people to like you, to do what you want. Rachel's bad at it, Taylor's decent, Kaiser is fantastic. Taunt is getting under the other person's skin, making them do something stupid. Rachel's bad at this, Imp is pretty good, and Tattletale alone was worth including this skill in the lineup, even if she uses her power to do so. Acrobatics covered running, jumping, endurance, etc. It would be called Althetics, except that I wanted something that could also cover sneaking if I had to. Genius covers planning, knowledge checks, and figuring out what another cape's powers are. It's called Genius because it fit the theme mostly- you can call it Intelligence if your table is more comfortable with that. It doesn't trump Thinkers, but it lead to some of the most audacious plans I heard at the table and was good fun. Power is the most versatile- it gets rolled whenever you use your power directly, to find out the mechanical strength of whatever you did. It's why Armsmaster is so much more dangerous than Kid Win, even if they aren't that much different powers-wise. It's a little bit of a god-stat depending on the power, but the two players I had who didn't max power did just fine with Brawl and Genius.
For aspects, I ran with three per character. One describing the power, one describing a relationship with a place or person, and one relating to the group. For a longer game (We only had five sessions, and knew that at the beginning) I would totally recommend going with more aspects, and letting the other aspects be a bit more freely decided. Offhand, five per character is typical- season to taste, essentially.
Stunts are unchanged from FATE Core, though they will likely be something to do with your Power. Wild Blue, a FATE setting that has powers somewhat resembling Worms, suggests having normal stunts and then two stunts specifically attached to your power, which would work very well I think. I ran with just two stunts per character for simplicity.
FATE has aspects, which worked nicely for covering the wide range of powers and allowing creative use of powers. Moral complexity is seldom a function of system, but nothing in FATE prevents it from occurring. (Didn't happen in my game, but I've seen it happen in different FATE games.) To hack FATE to do Worm, enforce each character having a power aspect and tweak the skill list to taste.
So, I mentioned above that Dogs in the Vineyard is a game that strongly encourages complex moral situations. It has some really strong points in its favour for a Worm game, as well as some really strong points against it. I do not have the experience running a Worm-style game with it that I have with FATE, but I have hacked DitV to do Star Wars, 1920s mobsters, and 3rd century centurions with an oracle, so I know the system is flexible. In fact, it's easy enough to change that I don't really need out outline how to hack it- If your players chose appropriate traits, and you changed the functions of Elements of Faith (or scrapped them entirely) and the Something's Wrong sections of the book, then you're done already.
Wide Variety of Powers
Dogs can 100% handle whatever crazy powers you can come up with. In fact, most Worm powers are a step down on the weirdness handle from your average Dogs game, though they are a little more explicit. See, Dogs in the Vineyard characters have three parts to them- Stats, Traits, and Relationships. Traits are... whatever you want them to be. In vanilla DitV, common traits include "I'm God's Watchdog" "Good shot with a gun" or "Solomn Attitude." Less common but completely legal traits I've used were "The old man across the river" "Bullet in my brainpan" and "I see dead people." See, each trait has dice associated with it (So, "I'm God's Watchdog, 3d6" or "Bullet in my brainpan, 2D8") which come into play when that trait comes up, and is more or less helpful depending on how many dice are associated with it. I could see making Taylor with "I control bugs, 3d8" and "I want to be a superhero, 2d4" for example. The ease with which Dogs can accommodate any kind of powerset actually would beat out FATE. (Burnscar, for example. Summing up what she does in one aspect might be hard, but in DitV you could have three traits- "Firestarter, 3d10" "Flame Teleporter, 2D8" and "My power makes me a sociopath, 1D4.") Heck, you could probably have half your players powered, half unpowered without affecting how much each can contribute.
The game is built with moral qualms at its center. The base game is about being given a gun and a book of scripture, and going into towns filled with crime and sin and trying to solve whatever problem is plaguing it before demons come and kill us all. It's about having power over people, and trying to exercise it well. And just as often, it's about people rejecting your power, and you having to decide how far you'll go to make them recognize it. The escalation system (which I'll get to in a moment) is one of the most important set of tactical choices offering far more mechanical power in exchange for making the fight increasingly deadly. Your character's traits give them a moral center, one that sometimes shifts as you try to stick up for them and get beaten down. Or as you win.
Cops and Robbers
Worm has a very interesting morality, and one of the odder quirks is the way the villains and heroes interact. I mentioned this when talking about FATE, where despite the rather alarming amount of firepower, there are very few casualties. DitV's escalation system makes perfect sense for this. Essentially, there are four levels of combat- Just talking, Physical but not fighting, Fighting hand to hand, and Fighting with Guns. Each level uses different stats, each and gives different penalties for losing. People can be operating at different levels (You're shooting at me, but I'm just talking) and people can escalate anytime it's their turn. Escalating usually gives you a huge boost, so you have a mechanical incentive to push things when you start losing. Those points where Taylor keeps thinking "Dang, this would be so much easier if I could use the poisonous insects"? That fits DitV perfectly.
The Godzilla Threshold
Here's the bad news. I don't remember a fight (Any fight, in the whole dang 30 arc tale) where the main character wasn't outgunned, and badly. Yes, that's just good storytelling, we expect to send our heroes against things ostensibly more powerful than them and to have them win. But in DitV, the PCs winning is not just expected, but darn near a foregone conclusion sometimes. The Jedi in the Vineyard game I ran a few years back had a session where the PCs decided to screw the light side, they were going to wipe out the city of Mos Eisley. And... they did. Handily. This goes back to the moral complexity- Any fight where all your dogs are present as full combatants and firmly on the same side is going to be a short one. Unfortunately, the combat system lacks a lot of tactical granularity. I would love to watch some Capes in the Vineyard try to deal with Coil, to look after territory and settle disputes, or to attack a dinner party full of capes while one of them is double dealing. On the other hand, I have no idea how I'd run an Endbringer assault. Maybe that's a lack of creativity on my part, but as far as I can tell, every big giant brawl would be me rolling a metric crapton of dice, the players rolling their dice and trying to bring in traits. It wouldn't be boring exactly, there would be lots of daring do and headscratching to bring on every trait they had, and I can see them grimacing over whatever fallout they got hit with. But no holds bared, us against them, center of attention battles is not a strength of the system and that's a fact. Now, fights with people who you really shouldn't kill but could, fights where all of the PCs are seriously trying to decide which side they're on or where two PCs disagree, those are closer to the strengths of DitV. Merchants good, Leviathan bad, Slaughterhouse Nine good. Anything after Arc 26? Hrm.
Thinking about it, if I was going to run an Endbringer or an Extinction scenario, I would make the fight really about the means used to stop it. Have someone offer a method of fighting it which you expect people to find reprehensible, and run that conflict while the Endbringer goes on in the background. The whole Khepri situation would work decently in Dogs, fighting the little battles to find a (Highly morally suspect) answer to something that can mop the floor with you if it every gets around to it.
A word to the wise
Little bit of cleanup. First, reiterating that moral complexity is absolutely something that you need to get signoff from your players to do. It can't be forced on them. If they go into this in the mood of a typical "Kick in the door" party, it won't work and everyone is going to end up confused and frustrated. Second, Dogs is a fantastic narrative game, and a lousy simulationist and gamist experience. Know your group. Lastly, DitV's default assumption is that you're traveling from town to town each session. You can stick to that (The Travelers and Faultline's Crew do this) but if you want to try sticking around like the Undersiders do, I'd suggest making a point to make the city big enough to have multiple pots brewing at once. Go through city creation multiple times, layering problem on top of problem so they can't solve everything in one go. Use stuff like Endbringer attacks or other large scale catastrophe's to hit the reset button once you run out of broken stuff for them to fix.
I would highly recommend the Dogs in the Vineyard approach to anyone who finds FATE too crunchy, any group with a seriously devious DM and players who love to make hard choices that don't have numbers attached to them, and anyone who wants to make narrative use of the truly oddball powers and doesn't mind if they are mechanically the same as every other power. I would recommend any Gamists avoid it like the plague, and anyone who needs the whole party to always act together may be uncomfortable at how often the DM will set them at odds with each other.