Keep the full cheek snaffle if you are happy with it. Perfectly good english bit (actually more an english bit than western).
As for the biting it sounds like he's challenging you. Major lack of respect and a very dangerous issue. What do you do to discipline him when he bites? When he attacks afterwards (what is he doing, coming at you with teeth barred? rearing and striking at you?)? When does he normally do this (tied for grooming, outside, leading, when he's loose)?
You might want to consider having a trusted professional help you with this as they can read your horse (they are there) and can hopefully 1) discipline him enough if he does it to them which will lay good solid precedent for you disciplining him 2) show you how to effectively discipline him to nip this in the bud so it doesn't happen again.