I use a Resource Pack that changes the look of the Inventory but when opening the Cosmetic Armor menu it changes back to the old style.
Is there a way for the mod to use the Inventory textures added by Resource Packs? It's super jaring without that happening.

It is up to resourcepack creator to provide compatibility for this mod.
Or you could just do it yourself, if the resourcepack allows.

I'd love to but how would I do that?

Make an addon resourcepack that modifies assets/cosmeticarmorreworked/textures/gui/cosarmorinventory.png to match the resourcepack you are using.

Oh okay. But that does mean I have to combine the resource pack with the Cosmetic Armor menu manually, right?

Yes, if you want to make them look matched.
You don't need to edit the resourcepack or the mod file, just create a new resourcepack.

How do I access the file for the Cosmetic Armor UI?
That way I could more easily just "combine" the two images.

https://github.com/zlainsama/CosmeticArmorReworked/blob/master/src/main/resources/assets/cosmeticarmorreworked/textures/gui/cosarmorinventory.png