Why don’t you rent a travel case from a music store? Or perhaps his music teacher has one he can borrow. My son checks his cello all the time - he has a flight case that fits over his BAM case. But he doesn’t have a particularly expensive cello. The one time (in over 14 years of flying), on Southwest, where there was damage - our homeowner’s personal articles insurance paid fully for the repair.
Also, if you expect your son to be attending college a plane flight away - it would be worth it to purchase a travel case.
However, if you want to buy a seat - it needs to be in the front row of the cabin. And they give you an extension seat belt for it.