I've seen a few posts about crow collars, which I assume is some sort of contraption that fits around their necks and keeps them from being comfortable enough to stretch out and let one fly. I've never read them, so I'm not positive, but it might be worth a search (and certainly seems much less invasive than surgery).
Are you legally allowed to have the rooster where you live? If so, then why worry about it? I mean, I can understand being a courteous neighbor and all, but I'm sure they could hold their classes inside or another part of their yard if its that much of a problem. It sounds to me like you're already going out of your way to keep him quiet. Not too mention, it could be an excellent opportunity for the neighbors students to learn about livestock, where eggs come from, things like that. Maybe you could invite them over to meet your rooster and let them take a tour of your coop (field trip, anyone?) and learn about chickens in general?
Anyway, good luck to you, I hope you find a solution that makes everyone happy!