His Mementos requests are all that's required- you don't even need to boost relationship with him. If you haven't done his request, you can't rank up his confidant. 10 is just like the rest- run the request, and it's a freebie.
Mishima is the worst for a lot of reasons, not the least of which is the fact that his confidant eats up over 20 time slots if you don't have Kawakami's maxed.