I'll share my experiences with a bolty Arabian mare I've had for 4 years. I don't know if any of it will apply. She has never offered to buck under me.
1 - Avoiding the bolt. Mia will give warning signs, but learning to recognize them and respond in an effective manner is a challenge. I've spent a lot of time during the last 3 months walking her around the neighborhood by herself, and it has given me some practice in noticing when she gets more tense.
It is a balancing act. If I ignore her nervousness and make her push on, she'll eventually explode. If I clamp down too quickly, she becomes angry and frustrated. If, when she is starting to get nervous, I redirect her attention without being a bully, she'll reconnect with me and calm down.
A - Disengages. If I've waited too long and she is REALLY getting tense, disengages will stop her, at least for a time. However, in walking her, I've noticed she resents more than 1-2 disengage requests. It can be good to do, but a single disengage needs to be followed by something else.
B - Circles. Weaves work better with her. If she is getting nervous and I push her to start doing a lot of circles, she gets worse. Weaves allow her to take a couple of steps forward, followed by a 30 deg turn, few steps, 30 deg turn, etc.
A big circle is better for her than small ones. The trainer and I think that is because she isn't very flexible side-side, and small circles affect her balance. That worries her, and makes her worse.
C - Bit cues. These work well if she is mildly nervous. Just stop her and ask her to bend her neck (not swivel her head), and do some head down cues. A minute of this can do wonders for calming her down.
D - Breathing slow. You, not her. Breath in audibly, hold for 2 seconds, audibly release. It sounds weird, but I've seen her calm on a lunge line doing this, and same from the saddle.
E - Audible cues. She likes hearing her name, and she knows what "Easy" means (calm down or slow down). In a bolt, I've had more luck with calling her name softly than anything else. Our gelding likes it when my daughter sings...haven't tried that on Mia.
F- Restraining bit. Haven't tried it on Mia yet (we've been going thru a few months of retraining with her). Have tried it with my gelding when he gets too worked up in anticipation of cantering, and it works better than turns when he is rushing around with his head up acting like an idiot.
As the right shoulder goes forward, just light pressure on the right rein. Not pulling back, just not moving it forwards with the shoulder. You can practice it at a walk and trot too. Just asking him, as he moves his front leg forward, to not move it so far forward. Then left rein. Repeat. Basically telling him, "not so far with the right, left, right, left, right..."
With Trooper, he'll go from rushing and tossing his head to a slow jog in about 5-6 paces. He responds better to that than turns, etc.
2 -In the bolt.
A - Turns to slow her down. Ummm....maybe. It takes room. Ask her to turn too quickly, and you set her up for a fall. A gentle turn, slowly getting smaller works if room is available.
B - Pulley stop. In my experience with Mia, it will stop her - for maybe 2 seconds. But she then starts again a bit slower than before.
C - If in doubt, ride it out. There are exceptions - cliffs, heavy traffic - but in an arena, it is usually possible to ride it out. Focus on keeping your knees apart, keep your balance, and just ride your horse. Mia makes me nervous because I have seen her fall at a full gallop with no one on her back...but she has yet to fall with me. Squeezing with your legs (knees!) will prolong the bolt...after all, the horse thinks you are either still scared, or asking to run. Stay in rhythm, knees loose, toes up, enjoy the ride...no, in honesty I find it darn hard to do as well.
But my best success at stopping a bolt has come when I make a conscious effort to force my knees wider apart, keep with the horse's rhythm, then gently call her name.
I really don't know if any of this will help your situation or not. Of what I've written, the best advice may be to walk your horse out & around, and let him teach you how to calm him when he is getting nervous. If you can't do it from the ground, you won't be able to on his back.