That is why I asked.
I went to a hackamore or just a halter because he rides so much better in them.
He may ride better, but he sure does not ride well.
I would probably try a bit just by hanging one in his mouth and letting him live with it several hours a day for several days. Some horse with badly made mouths will adjust to one and some don't. If he never accepts a bit, I would get a Dr. Cooks bitless bridle. They let you teach 'give' and flexion and 'bend' to a horse without a bit. I do not think they work as well at a bit, but they are a lot better than a halter, bosal or hackamore. They are better than a side-pull on most horses.
Then, I would teach this horse to 'give me his head'. This maneuver is commonly known as a 'one rein stop'. Then, you will at least have a way to stop any bad behavior cold in his tracks. It really does work, but has to be thoroughly taught before you go out and ride.
Then, I would start this horse from scratch. I would start with the groundwork and go up through all of the basic riding techniques. There are horrible huge holes in his training. He only accepts what he wants to accept because the means of teaching him to accept anything beyond those boundaries has never been taught to him.
If you wanted someone's program to follow, Clinton Anderson's would probably be the most complete and easiest to follow. The things that he already knows will work out very quickly. The holes will show up and you will be able to work on them until they are fixed.
When a horse has been taught to stop, go, bend, follow its nose and in general, accepts going anywhere and any way a handler want him to go on the ground, the same things work out very quickly on their backs.
Doing anything less is going to leave large gaps in training and control and will cause problems that can be very dangerous.