Hmmm I don't know if I'd call Chad a bully. He does use his physical size to try to intimidate people but I think his motivation is more out of desperation. OK this is a theory, but try looking at events through this interpretation and see if it makes sense.
Chad's whole position at B&R is built on a house of cards. He has a secret gay lover (never mind who for the moment) and is President of the Jocks. To maintain an illusion he keeps Sage, the President of the party girl sorority (who happens to have a crush on him), as his girlfriend. He has to know that if anyone ever made a serious play at Sage then he would not be able to compete, so he uses his only asset, his imposing physical size, to intimidate any potential rivals. Along comes the MC who is attracted to Sage and is not intimidated by Chad. This is his worst nightmare and as things crumble around him he becomes increasingly desperate to hold it all together.
This is strictly theory and guess work, but I think it may sketch out a plausible reason for Chad's behavior.