Okay, I think I might of figured a few ways for the University Athletics program to actually work. I'm using Athletics program as a sweeping term for all the various sports teams. Examples being swim team, volleyball team, soft ball team etc. The idea is that the program is divided up into different squads, or teams which compete against each other. This would be if each fraternity had there own team for any given sport. Another idea is that the dorms are divided into sections, and each section has there own teams under the Athletics program. For Example floors 1-10 would form one team, while floors 11-20 formed another team. You may also have certain floors only have a team in a single sport, instead of having more teams in multiple sports. This can also mean that visiting team outside of shokushu can be a bunch of disguised monsters.

Well, there isn't really any competitive sports in Shokushu, because of it's remoteness, and the involved costs. There are things involving the student body, but to be honest, anything involving monsters won't take place during said event, because most of the students there are off-limits to the monsters, and monsters cannot allow themselves to be seen there. Besides, sports are not exactly a focus at SHS, and your idea is just way too big for it.