For jumping, how about a pelham? Very similar to a tom thumb, but you can also use a snaffle rein if you want.
I (gasp) have been known to direct rein in curb bits, including tom thumbs. Gaited people do this a lot, so it is no unheard of. I just trail ride, but I do have friends from the gaited world, and they direct rein in a curb. Gently of course. I would rather ride with finesse in a curb than have to pull a horse around in a snaffle. But, it could be hard to ride with finesse in a curb while jumping, so I understand that too. A jointed pelham, maybe with the roundings to make it one-reined, would be the best suggestion I can come up with.
(Donning flame suit now.)