Since she's so good for you under saddle, maybe try a side pull? I just got one for my gelding and he's better than with a bit. The one I got is just a basic rope halter with pull rings. He was pretty good anyway, but he'd pull against it at the trot sometimes and I'd have to really work at getting him to maintain the pace I chose with a loose rein. I've only used the side pull twice, but so far he seems to be steadier at the trot. I really just bought it to see how he would do since I like using just as much "stuff" as necessary, but so far I don't see any reason to go back to the bit.
Maybe try riding her in a halter and see if she's suddenly out of control. If she's fine, which it sounds like she likely will be, you might not need a bit less bridle or bosal — a rope or leather side pull might be plenty.