I'm one of the few people on the forum who think a bit change CAN be helpful. In theory, every horse could be trained to go right in a snaffle, and every horse could be trained to go right bitless or even bridleless - but most of us do not have the time and facilities to train for that.
Some horses figure out how to ignore a snaffle. There is a reason the term 'bit in the teeth' meant an out of control person. No bit can FORCE a horse to stop during a full-up bolt, but that doesn't mean all bits can be as easily ignored. If your horse stretches her head out level when running, then the snaffle will just slide back against her clenched molars and do...nothing. The design of a snaffle means it is about as effective as a Q-Tip in the horse's mouth.
Some horses never figure that out. Some do. My mare has. I rode her in the arena in a snaffle a couple of days ago, and her excellent stops became sloppy. She considers a snaffle to be advice, and a sign I'm not serious. If she is in the mood, she'll behave perfectly in one, and she'll also behave perfectly in a halter. IF she feels like it.
A wonder bit is basically a western version of this bit:
The bottom two rings are shaped differently, but are essentially in the same location.
A leverage bit applies pressure to the poll as well as the mouth. The horse can clench its teeth, but that only stops the pain in the mouth. The pressure on the poll remains.
As with any pain we use for a cue, it becomes up to the horse to decide when she obeys, or if at all. Crops exist because some horses will ignore any amount of human kicking. When we say we don't use pain to communicate with a horse, we are only partially right. My cue to go faster goes from a click sound, to a light squeeze of the calves, to a harder squeeze, to a tap of the heels, and on up to a swat with a leather strap. About 90% of the time, my horse responds at the click with my mouth. Another 9% of the time she'll respond to a firm squeeze. But she also knows I can do more, if needed.
And if she is actually AFRAID of something in front of us? Nothing will make her go forward. She won't rear, thank goodness, but she can darn near trot in reverse no matter what I do - if it is important enough to her!
Same with a bit. She can slow from my seat cue before I move the reins an eight of an inch. Or she can go up the scale of intensity. In a snaffle, the most force I can apply is with a pulley stop, darn near ripping her face off. With a leveraged bit, I can apply pressure to the poll. With a curb, teh curb strap can add pressure on the jaw as well. She can run thru it as well, if she really wants, but she cannot make the pressure go away independently of my removing it.
A bit like an elevator bit or Wonder bit or a full up western curb bit of some style can be used to teach stopping to a horse who knows she can ignore a snaffle. It still requires some training for the horse. It is not just something you toss on a horse and have an instant solution.
I'm currently using this bit on my mare:
She responds better - faster, more willingly, and with a more relaxed back while just riding around - than in either the elevator bit or a snaffle. I've been experimenting with a 3-piece mouthpiece curb with a roller in the center. Sometimes she plays with the roller and relaxes, and sometimes she does not. The jury is still out on that bit.
I get frustrated when people say, "It is a training problem". Yeah, and reins indicate a training problem, since a perfect horse will ride just fine without them. I have two geldings who use snaffles - any snaffle. But my mare gets excited about going fast. If there is another horse there, she'll consider it a race. And she does NOT want to lose! I've also ridden a lesson horse who would not trot unless you had a crop. You didn't need to USE the crop. He just needed to know it was an option. As a 180 lb guy, I couldn't kick him hard enough to get even a little jog out of him. With a crop in one hand, he would trot with just a light squeeze of the calves.
Since there is a trainer there, ask if you can get some help teaching the horse about a wonder bit, or elevator bit, or a Billy Allen curb similar to mine. It took my mare about 3 rides to figure things out. We have also practiced our stops since last winter, using the advice in this video by Larry Trocha. The combination of a different style of bit AND training has made a huge difference in how she responds:
Good luck to you & your horse, whatever you decide!