This combination of those four factors has completely changed the flow of combat.
This is just a function of level in 3.5. The game changes massively from 1st level to 20th level, and you are much closer to the latter than you are to the former. It is normal and expected that the flow of combat has completely changed at this point.
In particular, invisibility and flight have been options for a long, long time. Anything and everything that is supposed to be a challenge at this level should have a way to deal with those. Unfortunately, the CR rules don’t actually accomplish that much – even the tarrasque notoriously has no way to handle either. So this is a failing of the 3.5 rules that supposedly-high-level characters and creatures can be shut down by such effects, because on a meta level these effects are expected. Flight, in particular, is considered absolutely-must-have-for-all-characters by around 10th, much less 15th.
Ultimately, this rogue is relying on invisibility and flight, which have counters, and precision damage, which many things are immune to, and his damage is only 200 per round – which is not actually that impressive at this level.
So honestly, the problem here is not that the rogue is too good. Greater Manyshot and splitting is a decent way to optimize ranged damage, but ranged-damage rogues have difficulty ensuring that Sneak Attack applies, since they cannot take advantage of flanking. His access to greater invisibility is only 1/day, and he absolutely needs it to do his thing. Anything that negates invisibility or is immune to critical hits, or simply anything he has to fight after he’s used up his ring, is basically immune to his entire character.
The real problems that I see are:
The rogue is an extremely binary character. Either he gets to be invisible, and it works, and he can deal decent damage with impunity, or his invisibility doesn’t work, he does barely any damage and is at considerable risk.
The rest of the party is not keeping up, particularly the fighter
You could, in theory, “solve” this by nerfing the rogue and continuing to try to play 3.5 as it gets into the highest levels, as if it still plays like 3.5 at lower levels. I think this is a losing battle, and don’t think it’s a good idea. In future campaigns, however, it may be a good idea to look into something like E6: it does not appear that high-level 3.5 appeals to you, because this rogue is just the tip of the iceberg.
Anyway, better solutions.
The rogue is very binary
Active anti-invisibility effects
The best solution here is to give enemies solutions to invisibility that are active – that they have to choose to use. This gives the rogue some opportunity for some free shots, and there is the possibility of trying to evade the effect.
Examples here include invisibility purge and short-duration, expensive effects like true seeing that someone isn’t going to have all the time.
Any passive “invisibility just doesn’t work on me” is going to lead to the rogue basically sitting out of the combat. For example, any 15th-level character I built myself would have multiple options for flight, seeing invisibility, and would probably have heavy fortification anyway, and would find this rogue a complete non-threat. I do not suggest this here.
Nigh-epic Listen checks
Extremely high Listen checks might also be appropriate. As difficult as it is to accomplish, if someone has a chance to hear the rogue, then you have a possibly-interesting fight where the rogue has to be careful about when and where he takes his shots.
This is a particularly useful option as DM, because you can figure out what this person has to roll to figure out how hard the fight is: if he can only hear the rogue on a 16, that’s a 25% chance to hear the rogue – he might not even live long enough to hear. But if success is on a 6, that’s a 75% chance and rogue had better pick his shot very carefully.
Mixed group tactics
Another option is a mixed group: a necromancer is an obvious choice. The necromancer himself is vulnerable to Sneak Attack, but if he’s protected from the undead, the rogue is going to have a very hard time getting to him, and cannot do much of anything to the undead. Works equally well for a golem-master.
Conclusion: 3.5 has a lot of ways to completely wreck this rogue
Producing fights in which he doesn’t function at all is not hard. Producing fights where he functions, but not perfectly, is much harder. Good luck with that. I hope the suggestions above are useful.
The rest of the party is not keeping up, particularly the fighter
Ultimately, this is a separate question, but as I described above – this is what 15th-level 3.5 looks like. Actually, it’s rather on the low end of what 15th-level 3.5 can be. The rest of the party should also be where this rogue is.
So some quick ideas:
Flight for everyone
The first step here is, every 15th-level character should have flight. It’s just too game-changing, and having flight when someone else doesn’t is so absurdly asymmetrical, that it’s just crucial. (The alternative is to agree ahead of time that flight is just not going to be a part of the game, which I have often done because I hate the flight rules.) So sometime soon, I strongly recommend that the party be rewarded with a blessing, a set of magic items, whatever, that allows all of them to fly. The rogue can be included; his winged boots can be a little faster or better somehow, or just ensure that he’s got redundancy against losing one or the other. But everyone else needs to get into the air.
A typical fighter at this level has a two-handed weapon and Power Attack, and probably either enough attack bonus or something like Shock Trooper to mitigate the Power Attack penalties, so he’s looking at a minimum of +40 from Power Attack, and he should probably have around 26 Strength at a bare minimum, for +19 from that. +69 per attack, minimum of three attacks, easily over the rogue’s 200 damage. And he’s at an awkward level; next level he gets a fourth attack and so possibly raises that to around 265. The last few attacks might not hit, but that depends how he’s been building.
If focused on charging, throw a couple of multipliers on there, so we’re talking easily 400 to 800 damage.
Apparently the fighter isn’t seeing anything like that kind of damage. Well, a trip-based build might not; a trip-based build is much more about lockdown. At this level, a tripper might control a 30-ft. radius sphere around himself, where no one can move, or if you’re really trying, do anything, without getting knocked over and punished. Which is honestly a much more reliable tactic anyway, and a good choice.
But I don’t get the impression that this is what the fighter is doing. Without more details, it’s impossible to guess what’s going on, precisely, but some ideas:
An animated shield
Give your fighter-type an animated shield of some type as loot or as a reward or something. If he had already been using a shield, and a one-handed weapon, this allows upgrading to a two-handed weapon, for massively more damage. If he had not, and already had a two-handed weapon (or dual-wielding), he now gets the benefits of a shield, which can be significant when you put magical properties on it.
If necessary, houserule things and offer more blessings, magical items, and so on, to make sure his feats continue to work with this change-up.
A better Weapon Focus
If your fighter has taken Weapon Focus et al., unfortunately that’s a very-weak feat. I suggest this version of the feat, personally, as an improvement. You could expand on the concept: maybe Weapon Specialization allows you to pick a weapon from the first group you chose, and for that weapon you can get the benefits of some other group it’s in. Greater Weapon Focus/Specialization could allow you to do that more, possibly even getting the benefits of groups the weapon is not a part of.
If the player has taken Weapon Focus, Weapon Specialization, Greater Weapon Focus, and Greater Weapon Specialization (which I do not recommend), I strongly encourage that he also take Melee Weapon Mastery and then Weapon Supremacy at 18th. Those feats, from Player’s Handbook II, are not good when you consider all of their requirements, but if you have those requirements anyway, they are actually very solid feats.
Some kind of ability to actually control the battlefield or protect allies
3.5 has extremely few options in this vein, but the fighter sounds like he needs greater presence on the battlefield. Giving him options for forcing foes to deal with him – either directly through some kind of aggro mechanic, or indirectly by punishing them for ignoring him, would likely work very well.
Unfortunately, as I said, existing options for this are minimal. It may be an inquiry to pursue further, however.
Consider Tome of Battle’s warblade
Tome of Battle is one of the last books published for 3.5, and without a doubt the most tightly-designed. The three base classes in it are extremely close to one another in balance, something no other book can claim, and they are very fun, very competent, very well-designed classes. They are also Wizards’ attempt to fix its mistakes in Player’s Handbook, and present the fighter, monk, and paladin done right.
The warblade is basically fighter-as-it-should-have-been. It is also available completely free, legally, on Wizards’ website. The Tome of Battle maneuvers, too, are also free-and-legal on Wizards’ website. Between these two, you don’t even need Tome of Battle to play a warblade. The only thing you really need to know beyond these is that a warblade’s maximum maneuver-level scales the same as a wizard’s maximum spell-level, i.e. half his initiator level rounded up, and that unlike wizards, half your non-initiating levels count towards warblade initiator level.
So, for example, a fighter 16/warblade 1 has a warblade initiator level of 9, and thus can select 4th-level maneuvers. It’s very late for starting an initiating career, but unlike every other subsystem for 3.5, that can actually work. Alternatively, retconning some fighter levels for warblade levels can help a lot.