I would do four things:
1) Change to a short shanked three piece 'colt bit' with big 'teardrop' rein rings. You can ride them 2 handed just like a snaffle, they are mild, and you can use a running martingale with them. They have enough more control in them that you don't have to pull so much -- which does indeed harden a horse's mouth -- or actually mind. You can usually school in this little three piece bits and go back to a snaffle after the horse's 'mind' had been trained while it preserves his mouth.
2) TEACH him to do a proper 'one rein stop'. You cannot teach a 'one rein stop' when a horse is out of control. It needs to be taught at all three gaits when the horse is listening and schooling well. here is a link to an old post of mine that explains in detail how to do it. It works on any horse from a reining horse to a OTTB. How we teach a 'One Rein Stop'
Any time a horse starts to get out of control and stops listening to a rider, a horse will 'yield' instantly if it has been properly taught to do this. It needs to be taught at all three gaits and it will take several rides at home to do it right.
3) Teach this horse to lope circles. No horse is 'broke' until you can lope circles anywhere on him. YOU should be defining the circle and not a fence. You gain nothing loping around a round-pen or an arena. When you do, the horse and the fence are defining the path taken. YOU need to define the circle and that can only be done by loping circles in the open or in a corner of the arena. If you use the corner of an arena, use the corner farthest away from the gate or from where the horse would LIKE to be. Only then, does the ride have any meaning.
4) Then, AFTER THIS HORSE HAS BEEN TAUGHT TO DO A ONE REIN STOP and after this horse has learned to lope nice circles, go to schooling locations (not enter a show) and find a spot and lope circles until he is listening. It is called 'seasoning' a horse.
You are just 'stealing' rides on this horse hoping he does not blow up and dump you. You need to actually learn how to be in charge and need to teach him that you can stop him and 'turn him off' any time you need to. You cannot take any horse, let alone one with holes in its training, (and the OP has a horse with HUGE HOLES in its training) and expect them to ride well in a show situation. It is not even reasonable to expect a horse to ride decently under the circumstances the OP is describing.