My dwellers will bleed happiness and complain about a corpse in my central cafeteria. I've checked the dweller list and nobody's dead, and I can't delete the room because it's at the center of my vault.

This is a known bug and the only way to fix it right now is to destroy the room.

If you build your elevators where they connect all the way down your vault you should be able to destroy any room without destroying any others or at most 1 extra room if you built them to full size so if your elevators do not work like that right now it may be more expensive and something for future revamp on your vault.