Haven't played Terraria in a while, but this is a good opportunity for me to put a good opinion on an upgrade system. It only depends on how much effort you're willing to put into the mod, but here you go:
When I think about an upgrade system, I think about a tiered system that grows with time so you need to constantly acquire new items. The most interesting approach would be that you need to craft the item rather than kill mobs to get the upgrade stone. Drops get annoying really quickly and can be abused if you have patience. If you need to craft the item, it sits on a "You gotta go pick these items and work for it," kinda deal.
So I'd suggest that that the player should craft the crafting materials in a tiered manner, starting with common everyday materials, like wood. So say you use 20 wood and 5 silk to create a "White-Tier Upgrade Slate" or something to that effect, which allows you to upgrade items which ORIGINAL rarity are White.
Then, to get to the next tier you'd put in some Copper/Tin and Iron/Lead to increase the Slate's rarity to Blue, to upgrade blue items. Then some Silver/Tungsten and Gold/Platinum to upgrade to the next tier, being Green, and so on. Of course, these are just coarse examples, you can use other more interesting items to make the upgrade item; I just used easy examples.
Of course, so it isn't a hassle, you'd be able to craft each slate individually using a slightly more expensive recipe instead of upgrading your slate, so you're not harvesting wood right before the Moon Lord to make some high tier slates.
When you get to around a third of the way (say +3/+4 or so if the cap is +10), the item has a small chance to decrease its upgrade grade. When you get around two-thirds of the way (maybe around +7/+8 or so), it has a chance to downgrade twice in a row. You can think of some other drawbacks the higher you upgrade, these are just examples.
Next, when it comes to the upgrading process itself, the slate that matches the item's rarity would have an initial 100% chance to upgrade the item, decreasing over a steadily rate the more you upgrade it, depending on how many upgrades you want to set as the ceiling. If you use the next tier, however, the chances are higher. For example, if you use a Blue-Tier Slate to upgrade a White-Tier item, the chance would be naturally higher.
A small problem shows up when you talk about Rainbow Rarity, as it's hard to balance out an item to upgrade any Rainbow-Rarity item. I'd probably do something where Rainbow-Rarities can use any Slate with a small chance of success, that increases periodically as the Slates get better, so to get a +10 Rainbow-Rarity you'd need to progress naturally through the game since they're basically equal in how useful they are.
Catalysts would be the last thing to talk about. Probably use Gems as Catalysts with different effects:
Ruby: Small chance to not consume an upgrade Slate.
Sapphire: Prevents upgrade grade decrease.
Emerald: Increase the chance of the upgrade working.
Amethyst: Small chance to upgrade the item twice in a row (half the chance to upgrade once).
Topaz: Allows a lower tier slate (one tier) to be used instead of an on tier one.
Diamond: Decreases the change of the upgrade working, but you don't use slates (upgrade can still fail or get degrade).
Amber: Increases the chance of the upgrade working by a lot (never to 100%), but if it fails the item resets back to +0.
Again these are just examples. You can probably figure out better ways of doing these too.
Also, I think that doubling the damage is way too much. Perhaps 25% tops would be best, with some small upgrades, up to like 5%-ish to all the other stats. Armor could probably get up to 33% more defense and +10%-ish its other stats (when applicable). Accessories would be hard, probably either some small defense amount, some enhanced effect or increasing their prefix bonus by 50%-ish percent when +10.
Lastly, mod support is a big thing. Some mods introduce new grades after Purple, so it'd be nice for a slate that works with those or something. The more compatible and interactable this is with other mods, the better it'll be since it seems the majority of the player-base uses all the big mods out there (myself included when I used to play frequently).
Hopefully, I've given you some ideas. It really only depends on how much effort you want to put on this mod. Would be extremely nice to have at least 1 good upgrade mod out there that isn't super exploitable. Thanks for the work you've already put in, btw.
Best of luck!