Eh...it COULD be something he picked up as a lesson horse. Who knows...there are quite enough sour lesson horses out there to make it a possibility.
I suggest taking it slow; a full cheek snaffle is perfectly fine, you can try another if you'd like, but I'd stick with that level of bit at this point. Work on getting him to flex his neck, to give to the bit smoothly and softly. Do figures, do lateral work. If he doesn't want to stop, don't fret about it, just keep him on a small circle till he does listen. Get rid of the tension and he will be more amenable to the aids. He'll be less likely to lean on the bit and just plow along.
Anyway, that's what we do with the ottb's, so that would be my first strategy. If he's still uncomfortable with the bit after a couple of sessions, I might look into trying something else.