What is the strongest bit you can use in horse riding?
...without damaging or hurting the horses mouth?
Any bit can hurt/ damage a horses mouth depending on the riders hands. I've seen snaffled gouge and hackamores cause nosebleeds. If you're having an issue, it's probably better to re-school the horse.
any bit can injure your horse if your horse is not responding to you a stronger bit isn't going to help I would recomend that you retrain your horse and gain his respect.