I agree with SD.
And, I taught my mare to neck rein in a snaffle. She seems to understand the cues better in a bitless bridle but she understood the concept already from the snaffle.
I think you need to go back to basics with him and retrain him to stop and do all the other stuff you're having difficulty with, in a snaffle. It might be helpful to teach him some voice commands on the ground then transfer them to the saddle.
For instance, when I got my mare she had a great stop but she couldn't stand still for anything. I trained her on the ground to stop and not move when I say "ho" until I say "ok". I would say "ho", stop my own movement and have her back up 3-4 steps then stand for a moment (eventually I started taking a step away from her and saying "ahhh" when she moved etc). She picked it up pretty quickly and now it's SO nice because I can tell her "ho" and trust that she'll stay in place until I tell her she can move.
Now I use it in the saddle and she's great about stopping and standing still whenever and wherever I need her to. And even when she decides to tune me out and be a jerk I can say "ho" to her and she'll stop dead.
The only thing I have to be careful about is to not say it without making her stop if she doesn't and I have to be careful to remember to say "ok" because I don't want her to learn that she can just stop then walk off when I say "ho".
A stronger bit isn't going to solve anything, it might even make it worse. I wish you luck! =)
Fabio - 13 year old Arabian/Lipizzan gelding
Hazel - 14 year old Angora goat
Atticus - 4 year old LaMancha/Alpine cross goat
Rest peacefully, Lacey.