I have a character that is multiple people sharing a body, and I want to be able to have multiple outfits for each person, but I'm running up against the 5 outfits per category limit

I think the problem is that you only don't see more than 3 outfits in Stylist, but if you add another, it will be there on the sim, just change into the third fourth or fifth befor you enter Stylist.

I have a sim with 7 outfits in each category, so I can transfer clothing to others if I don't want to style a new one (or use dresser with change random). You can save the sim to the bin in Stylist (Saved Sims folder), to save the look and all outfits
this saved sim copy includes all outfit levels and you can simply bring any sim in
- CAS during game to exchange the Sim with your SavedSim copy

In game you can check via Mastercontroller -> Sim > General -> Outfit
You can also transfer outfit to another sim (same age gender)
Intermediate -> transfer clothing
choose add item index or exchange existing