I would look at re-training before getting a harsher bit. I don't believe that a harsher bit solves anything and it would only be a matter of time until the same problems turned up again. Try riding him on a loose rein and when he starts to speed up, either stop and back him or turn in a one-rein stop (switch directions for stop often). This should help him to control his speed better. When I have a horse that is not responding to the bit and trying to run through it, I will get them on a flat a long way from home. I let them jog a little and then ask for a stop. When they stop, I release pressure and ask for a jog again. I do this often and before long, they are responsive to even the slightest cue to stop. When they have that down, I start asking for a back after each stop. This will get them stopping harder and faster. I also keep pressure on until they give their head and flex at the poll even the slightest bit is rewarded with release of pressure. It will take a lot of time and patience to work through these issues and I would make sure that you have control with bit and legs before you try working on direct contact because it sounds like to me, he wasn't properly introduced to it and has started nosing through the bit. The best option is to start from the beginning and retrain him. I wish that there was an easy fix, but with this kind of thing, most of the time there isn't. Good luck.
Also, I don't think that a regular snaffle is that bad. That is what I use on all my young horses. Since you already have one, that is what I would stick with.