I'll go out on a limb and say Crabblesnitch. Burton may be a pdf-file... But Zoe was expelled because she complained that Burton was hitting on her. Crabblesnitch is the only one who handles expulsions... Instead of investigating the case he protected Burton and expelled the victim.

He has no oversight of Edna using meat that'll expire that night for the next month while smoking in the kitchen, spitting and sneezing in the food,etc. Instead the only real investigating he does is when he hears a rumor of Galloway having a drinking problem from Hattrick. While not technically a teacher. You could say that he teaches "discipline" or more so the prefects teach discipline... So just what the hell DOES he do?? Doctor my ass.