I don't like happy mouths in general because I find them too stiff, I like more flexible bits. Also not a lot of salivation and some horses will chew them up. Sure, you could try it, but you won't have any more brakes than you would in the little S.
If you go in to run him and he's acting like an idiot, walk, trot, lope the pattern. Do NOT run him if you aren't sure what he's going to do. Good way to get both of you hurt. Maybe take him and exhibition/time only him. If he acts okay going in the gate, ask him to run, and if he gets goofy go ahead and collect him back to you slowly throughout the run so by the time he gets to the gate he's already going slow.
One thing is tricky about barrel horses. Some are idiots going in. My sweet, calm, little Selena is a freight train when I go in but she's listening. If you don't think your horse is going to listen, get her relaxing. Stand next to the gate while people are going in and when the gate opens, get off and loosen your cinch. Do those exhibitions and keep her chill. Once you get confident she is going to listen to you, let 'er go.
Have you ran her at home before? Does she stop? What happens when you ask her to stop from a faster lope?
Pssh.I didn't pick up the wrong lead
It's called a counter canter...
...A very advanced maneuver.