Whichever bit or bridle makes you feel that your horse is "listening" to you would be my choice. Of course, if there is discomfort, like a bit that hurts a hrose, he will be listening (thinking about) the pain. So nix that. But if no pain is part of this equation, ride him some and when you ask for a turn, watch his ears and eyes. Say you turn left. You pick up the left rein, and maybe put a slitght twist in hyour body. when you pick up that left rein, does his left ear swivel back some? Does his eye maybe roll back toward you a sec? Does he flex a tad at the jaw?
I am sure that when he disobeys you , his mind is going one way and your rein the other, so you get resistance and pull. In such an occasion you need to get his mind to the rein that is talking. See which one works better at this, the bit or the sidepull.