Go back to the smooth snaffle, correct training will get him to soften up. Lots of horses don't like a single jointed snaffle and will prefer a french link. Other horses are fussy with the cheek piece, I know quite a few horses that don't like loose ring snaffles as it moves around to much and prefer the stability of a D ring or even a baucher. Training comes first however. Switching to harsher bits doesn't work in the long run as over time you'll need to keep moving up as the horse learns to evade that. Even switching to a different kind of mild bit isn't going to show you much in the way of a result, training comes first. Then if you notice somethings could be better, try switching something up.
When switching bits have two bridles set up. Ride in your old set up for half the ride then the second bit the next half. This will eliminate any variables on your horses mood or behavior between days.
Training wise, start with lateral flexion and offer softness. Go with your horse if he moves his head around and the moment he softens and puts a little slack in the lead rope (I teach this on the ground first), release.
When teaching a horse to give to the bit, you always want to ask with the smallest amount of pressure. A horse will be more inclined to pull and resist if you ask harshly or can learn to be intimidated by the bit and 'hide' behind the bridle with bumping. On a horse that knows better, sure bump away if he's ignoring you, but you always want to start with the softest cue possible. Starting softly will teach a horse to respond to that, if you start with a heavier hand, the horse will never respond to less than that as he doesn't know to do so.