So the horse bucks as the daughter is trying to get him under control when he bolts, but goes perfectly for the trainer.
I wouldn't attribute it to saddle fit or bad hands. Sounds to me like he bucks when someone with a little less finesse clamps down and braces against him. Bucking's usually caused by discomfort in the back, and sometimes by a feeling that the rider is off balance. It's a natural response, get rid of what's there and get away from it. A lot of horses who buck with some riders won't buck with a better rider. I have a horse in training now who bucks whenever he 1) fells he's getting left behind, 2) spooks at something, 3) has just jumped his first or second jump of the day, 4) gets a little excited about running. It's almost an involuntary reaction to the tensing of his back muscles. I just try to avoid those situations and keep him supple. Successful bucking, ie dumping the rider, is bad for the training of these horses. They associate bucking with being free of the rider, and they learn to keep going and adding twists and turns until you hit the ground.
I sympathize. It's not a fun predicament.