Point blank, if you're going to show in dressage, your horse needs to be wearing a snaffle bit. Them's da rules. You could just ditch dressage and keep on pluggin' with the gag and the kimberwicke, which are not bad bits in and of themselves, but without a solid foundation of obedience and acceptance of the snaffle its unlikely that these harsher bits are being used as correctly and successfully as they could.
Yes, horses are horses, and none of them behave perfectly 100% of the time. Push button horses are made, not born, and that's the rider's job. Going back to Anebel's piggyback kid analogy, your horse might not fall down if you lean over too far to the left, but think about how you would accomodate a small child... you would lean to the right! It isn't easy to walk like that for any distance, let alone jog, run, or jump. After a while of carrying the kid, it's going to hurt! If you carry that kid for an hour every day, after a while you're body will have some damage from not moving the way it should under a heavy load. The best thing that any of us can do for our horses is to learn how to have a good seat and sit in such a way that we don't interfere with the horse's ability to move as he needs to to get his job done.
If you aren't "making static" in the saddle, the horse is freer to pick up on rein aids, and you shouldn't need a lot of metal in his mouth. That being said, it isn't the only fix. Horses aren't just hard in the mouth. Hardness in the mouth is a symptom of hardness somewhere else in his body. Fix the body and you fix the mouth. Rule #1 of dressage - the VERY LAST thing that its about is the head!!
My advice: Step 1: Find a snaffle that works for you. I ride my greenie in a single joint eggbutt with a copper mouth - pretty bare bones mechanically. Your horse might prefer a French link due to his specific mouth conformation. A full cheek can give you a little clearer of a "turn your head" cue, and a loose ring can keep the horse from hanging on the bit, but there are very mild effects. Step 2: If at all financially and geographically possible, find a dressage instructor/coach to help you sort things out. He/she will be very able to help you find the holes in your horse's foundation and fix them, as well as helping your riding and his training as you both progress.
If you're going for eventing, you may always want a little more bit in the XC phase, but I'll defer to the event riders on that point. Good dressage will help amazingly with your jumping, stadium and XC. Jumping in any form is just dressage with obstacles.
By the sound of it, this will be a very different way of riding for you, and you may have a lot of work retraining yourself and your horse. It's something that I'm working on, too. The good news is that it is very worth it. And I left out the "magic" part... the more you focus on what you are doing, instead of what your horse is doing, the fewer problems he'll have.