I'm not trying to be mean, but it doesn't sound like it's a bit problem, it sounds like it's a training problem. Most lesson horses don't have a lot of finesse (because all kids do is pull and haul on their mouths), but regardless, what you are describing is a training issue.
Put him in a snaffle (doesn't have to be a full-cheek... I usually use full-cheeks for baby's only, but no harm done!) and teach him to CORRECTLY give to bit pressure, and neck rein ALL the time, not just 'when he wants to'. Going to a bigger bit to get more control doesn't mean it's a bit problem, it means the horse doesn't respect a rider until there is physical pain involved--ie, he isn't trained well. It's also the reason why he doesn't stop. You say, 'whoa!' and he says, 'not unless you make me'.