Your trainer may be able to help you with your bridling issues as well. Perhaps something you are doing, is causing him to be able to toss or pull his head away while you are trying to bridle him.
As far as 'what' to ride him in, for barrels, you just about need a bit that the shanks move seperately from the mouthpeice, so I would either go back to the snaffle, or make sure you get a curb bit that has swivel cheeks on it. You could try a hackemore as well.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."