And the reason I'm not so keen on a snaffle with a running-through-the-bit barrel horse is that the tendency is going to be to puuuuuuull them to a stop when they fight. They're already bit-sour, and this can just compound the problem and toughen their mouths up even more. It's been my experience that they're far more apt to respond to the curb, as they've been trained in and are used to its pressure.
Of course, it's not like there's a definite right or wrong way to go about this. I've just had a fair amount of personal experience with this very thing and have found this method to be effective....but in the end, whatever works, works.