Especially if he's green, training is definitely the answer. Take things very slowly & easily, focussing on & reinforcing him relaxing & concentrating, rather than just going over the jumps. Also teaching him to do everything on line first, before attempting it mounted.
It's also possible that a different saddle, different position of the rider or something else could be causing discomfort that makes him more 'excitable' in this setting. Be sure to eliminate any physical causes.
When talking about the 'strength' or 'severity' of a bit, you're talking pain. If you need to use a bit 'strongly' you are attempting to control the horse through pain, with whatever bit. 'A little stronger' means the ability to create more pain. Other problems aside, if you're aiming to produce a happy partnership with the horse, that's not going to be a productive approach.
A single jointed snaffle can be quite severe, having the ability to act like a nutcracker on the tongue while gouging the roof of the mouth. This could become a problem with a horse that isn't trained well enough to only use the bit gently for communication rather than control. Perhaps this bit is creating or aggrevating the problems in this setting, while being fine in dressage where he's less impulsive. I would suggest training him first in a halter or a 'gentler' bit such as a double jointed or rubber one.