I would do groundwork training to encourage a more responsive stop before jumping to a bigger bit. Any horse can run through any bit, specially a draft cross, that's why you should TRAIN a horse rather than just use physical pressure to try and out-muscle them if you would. Get him on a lunge line, get a solid "WOAH" so that he slams on his breaks and backs a step before trying it in the saddle, get that "WOAH" to become habit for him.
For those emergencies where you NEED to stop NOW (which you should avoid, as you do not want to set yourself up to fail), I'd learn how to use a one-rein stop or learn how to get him to turn and give to the yields. A horse has to follow where their shoulder goes, so learn to move that for emergencies.