When you school, teach him that cross country is not exciting and that he still needs to listen. Try trotting fences and stopping after then backing up a few steps until he listens. Then repeat the exercise but cantering fences.
You may also want to try a kineton noseband. It will give you more leverage on your horse’s nose rather than utilizing a bit. And I think a stronger but such as an elevator or Pelham is a good idea. Someone above mentioned that harsh bits can ruin a horse’s mouth. The thing with bits is that any bit can ruin a horse’s mouth if it is used incorrectly, even a snaffle. Bits are a tool just like a whip or spur. You need to make sure not to hang on the horse’s mouth or seesaw the bit.
A hackamore bridle may be a good option if you are nervous about using a strong bit. Just remember that even though it is bit less, it is a very strong bridle that can hurt your horse’s nose if used incorrectly. You should discuss with your trainer and get his or her input and help with your decision.