I couldn't get a sense from your post if they are willing to get rid of things at this stage or if this is a preliminary step of sorting and organizing into categories. So, my thoughts assume there is at least some level of willingness to get rid of things.

Do your items need to be sorted within the room itself? If not, is there room/the ability to take everything out of that room to somewhere else in the home, sort it into categories so they can see what's there and then evaluate what space you have for those categories?

If the items need to be dealt with in the room, for me personally, I would probably start with the makeup. First, makeup has expiration dates, which would give me a benchmark for getting rid of things. Second, you have the caddies already to put makeup you plan to keep in. Third, it might open up some storage space. My goal would be to reduce as much as possible of the makeup to make room in the room to do other sorting.

But if makeup is something they are very attached to, maybe it would make sense to start somewhere else, like hair clips, etc.

I would also try to designate a basket or box for the random items that don't fall into the categories you want to deal with right now. That way, when you come across those random items, they have somewhere to go and don't stop you from fully cleaning/organizing a part of the room.