Trophies go in houses. Trophies are accessed through the tribute tab. Just add another tab to the tribute screen and you can access all trophies at one time. Problem solved.
I like the idea someone had in another thread, to be able to add the mods to 1 last level of the ts ear, or artisans prize, requiring you have all standard skill trophies or mods at 15%, then only the oddballs like tinkering poison and alchemy would have separate trophies
Another thing that would be cool is to leave the trophies as is, but add the ability to summon the common tools for that skill, like a hammer, needle, Jewelers eyeglass, etching tools, etc then can save THAT space in bags by not having to have them on you too. That said, I keep trophies on me all the time and have 2 bandoliers for equipping them, but I dump all my ingredients into housing after each session. Often I keep the tools too unless it's excessive(clay stamps and cutting tools can be but at least the cutting tools can be converted to each other)
As has been suggested, simply add a new level to each of the 'xx mastery' AA lines that is only buyable if you have a 7/7 trophy, that grants the % increase that the trophy grants. Then, we can drop our trophies in the bank, in out house, wherever as we wouldn't need to be switching them out. If you want to be nice, have each of these AAs summon a bag of the 'common' TS items. (I'd say make them norent versions but that would probably require duplicating all the existing recipes?)