Just to reinforce some things that have already been said; horses don't start bucking for no reason. The most common reason is some sort of physical pain.
Based on the way you describe your horse, which as a well-broken, good guy who is normally tolerant and forgiving (he even LOOKS like a good guy and a horse I would want in my barn, I would bet some serious money that it's some sort of physical issue that's causing the bucking.
Saddle fit is the obvious first thing to look at. Poor saddle fit is sometimes obvious but often very subtle and the effects can be cumulative.
If you really look at saddle fit and that doesn't appear to be the issue, I would look for some other physical issues. If canter departures appear to be an issue, I'd pay careful attention to his back and hocks.
If you don't have a chiropracter you trust, have a vet do a complete, thorough physical exam and see if he can find a source for the discomfort. Consider getting an equine massage therapist to work on him as well, she/he may discover a painful spot.
No reason not to have your trainer go ahead and school the horse, it's never a bad idea, but I would guess that she'll encounter some physical resistance as well. Maybe not as dramatic as the bucking, but I suspect it'll still be there.
If the horse bucks with her in the same way that he does with you and your daughter, that's another indication that it's physical rather than behavioral.
I understand that the bucking is frightening and disheartening, but there is not cause to give up on your otherwise lovely horse just yet.