Your horse sounds like a lovely boy. I really love Saddlebreds, especially those with that showy attitude and a lot of action. I personally would move to another trainer if yours is impatient and unwilling to work to correct problems in a horse, especially a horse his client obviously loves. I don't have a lot of tolerance for trainers who aren't willing or able to fix problems.
I would also have an equine veterinarian examine your horse. Equine chiropractors can play an important role in treating horses, but not all equine chiropractics have the same education and training. Some are very good, but some aren't. I would want an equine veterinarian to do a lameness examine, examine the horse's back and look at his hoof care.
If the veterinarian doesn't find anything, then I would want a trainer to look at you and your riding. Some horses are very sensitive to their riders and subtle shifts in their rider's weight and/or position can cause them to change leads. I have a horse of mine that won't walk a straight line for a novice - he's so responsible to seat aids and leg aids that a novice rider who is insecure gives him so many conflicting cues that he zig-zags all over the place. Have the trainer check your seat and your legs as you ride. You may find you've been inadvertently telling your horse to swap leads. You also might have the trainer ride your horse to see if he swaps leads for the trainer, too.
If both your horse and your riding check out, then I would work on building up his strength. Horses normally swap leads because either they have a physical problem (most common cause), their riders are asking them to or because they're out of shape and unable to canter long periods. Start slowly with trot work and very brief canter sessions and build up the length of riding/training sessions over time.
Once he's in good shape, give him longer canter sessions. If he continues swapping leads, then treat him like a young horse who doesn't really know how to canter. Ask for very brief canter sessions, bringing him back to a walk or trot before he has a chance to swap leads. When you start getting quality short canters, gradually extend the length of time you ask him to canter. If he does swap leads, bring him back to a walk and ask him to pick up the correct lead again.
Good luck with your boy! I think you are fortunate to have such a lovely horse and hope you are able to continue working with him.