I have several bits with that mouthpiece, and never knew they were called Billy Allens...thanks for the link!!!
I really like them, no horse ever "complains" about them, and even in uneducated hands, the horses seem to work well.
Should be a good step up from a snaffle, if he is running through it.....but I have seen barrel horses run through bicycle chain gag bits, so I agree, training is a good answer!!