You might just wrap some latex on the bit you have been using, to pad it a bit. Lots cheaper than buying a bunch of new bits that may not even work. I have heard a number of stories about some of the rubber bits failing after some use. Others get the rubber chewed to shreds.
I suggest just a couple latex wraps on the mouthpieces that lay on his bars. I cut the latex to fit the width of part i want covered, not the entire mouthpiece. This lets bit joints work easily when reins are applied. I pull the latex as I wrap it, to get stretch in making thinner layers, plus it sticks to itself better. Cut it off, then squeeze the latex for best adhesion to itself. Horse will probably foam up wearing newly wrapped bit, new latex seems to cause that. Everyone will think he is really using the bit well! Ha ha
Not sure how much you practice ride, but you sound like horse needs practice with bit contact. He is refusing to “give”, drop his head, when rein pressure is applied. Probably does not know how. I call it an “untrained mouth” over being sensitive. Practicing this giving and release in the arena, at a walk, will help him understand better. You take up reins, apply steady pressure but not halting. Then " throw reins away" when he drops head removing pressure even the tiniest bit! He learns that giving lets him get the REWARD of no mouth pressure! You have to watch carefully for the give, it can be a twitch or almost microscopic, to give him the loose rein reward.
It is a learned response, getting into self-carriage as you get better with that rein reward. He has no reason to give if the pressure never goes away. Do you use any vocal cues for stopping? Many horses are quite willing to help stop, when vocal cues let them know stopping is next! Not all halting with reins. He may already know Ho, from being led, not walking over the handler. Try using Ho with firm rein pull, for stopping. Then release reins as he stops, for the reward of his halt. Pat his neck if possible, to make it clear he did the right thing right then!
Good luck with him.