Any horse I've ever ridden I've ridden in a snaffle, whether the horse had a soft mouth or a strong mouth. Personally, I hate stronger bits. I don't see the point in putting a stronger bit in a horses mouth if the horse is already strong in the mouth, because chances are all your going to do is pull and pull on the horses mouth and cause the horses mouth to become even stronger. The only horse I have ridden in anything other then a snaffle was a Arab/TWH/TB cross that had a very sensitive mouth, but used a kimberwich bit. She had been trained in a Kimberwich bit and would not use any other type of bit, but was also dressage and would freak out if you pulled on her mouth the slightest bit. You didn't need to use your hands other then to have contact for her to come into frame, you mostly rode her with your body. Maybe you could just work on excersizes to make your horse listen to your body more rather then having to rely on your hands to stop/slow down your horse, and that will get your horse off the bit (not just taking the bit and being strong) and using her back end. Just a thought. Good luck