There are a ton of monsters that are immune to fear RAW. Does he expect you to never use undead against him?

This is not a case of a "bad GM", this is a case of a bad player. He's trying to coerce you into letting him stay unstoppable with his unbalanced 1-trick pony.

Personally, I rarely have fun DM'ing for a player that doesn't even question their own character's balance. They just want to "win" and their "roleplay" usually only involves them abusing NPCs in some way for their own gain or revenge.

I'd HIGHLY consider talking with him, either one-on-one or as a group if other people have concerns, about your thoughts on it. I've had to ask some players to leave before due to not reaching any agreements or compromises, because their presence made the game not only unfun, but stressful.