My first thought is that you should teach your horse a reliable stop on a “whoa,” which doesn’t involve the bit. No bit is going to stop your horse if he doesn’t know how to stop. Start on the ground in a halter, long lead and with a dressage whip or training stick. Walk forward with your horse at your shoulder with about a foot of slack in the lead. When you want to stop, first say, “whoa,” then stop walking and GENTLY tap him on the chest with the training stick. Don’t pull on the lead, let him walk into it. Praise him if he even slows down. Work on this, gradually giving him more and more lead rope, until he is stopping on your voice command and following your lead in stopping your own body. Then get on and walk forward as before loose reins (or even better do it in the halter as he now knows how to stop in the halter). Say, “whoa,” stop your body motion in the saddle and tap him on the chest with the training stick. Again, praise him and pet him on the neck when he does it right. Don’t practice over and over at the walk. I repeat DO NOT keep practicing. He now knows how to do this, so there is no need to keep teaching it to him.
Now do the same mounted exercise at the jog and lope starting slow and easy until he wants to stop on a “whoa” or at least gear down one gait. You should not be touching his head or mouth with the reins as there will be no need.
Once he’s reliably stopping from the lope, try walking your barrel pattern, and stopping at various places at barrels, between barrels and at the end of the pattern so he doesn’t know where he’s going to be asked to stop. Gradually quit asking him to stop on the pattern and ask him only to stop when he’s finished it. Increase the speed until you’re at racing speed. You should have a fantastic stopper who will go in any bit, a hack or no bit at all! I did this with a former racing Appenix QH who had no stop and wound up with the best stop in the barn completely bridleless. Good luck.