If you are going to use a rule, use it in full.
Looking closely on how Fate works, I don't see that giving to weapons a weapon rating is a problem, however I can see how giving to only 2 weapons a weapon rating could be.
The thing you cannot ignore is that everyone have an armor and a weapon, even if it is a natural armor (your body) and natural weapon (your fist, nail, teeth, head, etc), and all weapon and armor should have an armor rating. Most of these are defaulted to Zero, and this gives the impression that you don't have a armor or weapon, and this feeling can lead to people feeling cheated.
So here is a few things you can do about it
1. Do not use the Zero rating.
Change the default to 1. Every weapon and armor would default to 1, and the game would be as balanced as if the default was Zero, but now with everyone being acknoledge as holding a gun and having an armor. The Katana and Bat players should still have better scores, but that can be justified by having spent an aspect on it.
2. Rate every weapon individually.
The Haunted Katana, Adamantium Bat and the Neo-Baseball Armor aren't the only weapons and armor. You could give an armor rating for the biologic modifications and a weapon rating for unarmed strikes on the augmented cop, give pistols, submachine guns, energy pistols (for the space pirate if applicable) all of them with weapon and armor ratings. You could also discriminate lethal from non-lethal damage and resistance, and having separate stress tracks for them, just be careful to not make the system too complex.
3. Make player equipment superior from NPC equipment.
Instead of just changing the default to 1, you could give all players minimum armor and weapon rating as 2, and enemies NPCs minimum armor and weapon rating as 1, but give the NPCs 1 (or two) extra shifts. Your players characters are competent people and having a better equipment than the normal enemy minion can work as a good morale boost, just make sure that if an enemy is supposed to die in 3 hits or 1 lucky roll, it doesn't fall appart to 1 normal roll of the space pirate/augmented cop.
4. Give the weapons also a limited social aplicability.
The Haunted Katana most likely will have a spirit attached to it, and this spirit can do more than just harm, it could help the player in non-fight situations. The Mass-effect like armor could have an EDI like AI attached or remotely connected to it that would have the same function.
5. Treat non-combat encounters with weapon and armor ratings as well.
Your agumented cop have psychic habilities that would give him an edge on social enconters, your space pirate have spaceships that can give an edge on traveling/speed encounters and a pet that can give him an edge on stealth enconters. If you treat all encounters as having "weapons" (things that help you progress) and "armors" (things that help enemies counter your weapons) the "unfairness" may simply go away.
6. Don't use weapon or armor ratings, but enforce aspect invocations rules.
Everyone can create an advantage with a attack skill, but not everyone should be able to spend a fate point to get +2 on a attack roll based on aspects. If there is no way to make your players not feel cheated just don't give armor or weapon ratings, but remeber that is easier to justify being better at dealing damage because of a Haunted Katana, than because of having a Space Parrot.