Mmmmm......I have a different view on this
One, get that tie down off him. My horse was ridden in a tie down for years before I got him and I had SO MANY issues to work out! Tie downs, IMO, give the horse something to brace against, plus it forces a horse to carry his head in a position that he isn't WILLING to do. A tie down is a band aid for a much deeper cause of pain and/or unconfidence.
So here is what I would do. Make sure you saddle fits. Triple check it! Make sure it's in the correct position, behind the horse's scapula at MAXIMUM extension and make sure your saddle is then shimmed appropriately to make your saddle level. You dont' want it putting pressure on his withers. If your saddle is blocking his shoulder movement that will cause him to hollow his back and stick his head way up in the air.
Next, I would get a double jointed snaffle bit. I would suggest getting the JP by Korsteel Oval Mouth Copper Loose Ring snaffle. You could also use a KK snaffle. With this snaffle you then want to teach him to reach for the bit. Your horse sounds exactly like my horse used to be. Once I tought him the following exercise he COMPLETELY changed. I do not believe this is a "bad habit" or something he is doing out of snotty behavior, I believe this is a physical issue (might want to get a chiro out), along with saddle fit, and a confidence issue. A horse who is comfortable and confident WILL NOT hollow out like that.
So once you have the snaffle get on and start this from a halt. Start "combing" the reins through your hands with contact in his mouth. Use a flowing, rhythmic feel and allow the reins to slide easily through your hands. DO NOT block him. If he starts backing up hang with him and even put his butt up against a fence. This is a hand-over-hand motion. What you want him to do is to reach for the bit and stretch down. When he does release the reins completely and rub him. HERE IS THE KEY......the higher he puts his head up the more drag you put in the reins!
You are insisting that he keep his head lower. What I mean by more drag is that you don't allow the reins to slide as easily through your hands BUT YOU DO NOT PULL ON HIS MOUTH. If you pull, his head will come up. Once he is okay at the halt then go to a walk, and then a trot. DO NOT do this at the canter. Your goal is to have him stretch waaaaaay down. Make sure you don't block him or pull on his mouth! If you are confused by this let me know and I will explain better.
With my horse this changed his body physically. His topline improved dramatically. And the more you do that exercise the more you will see your horse carry himself differently. Even on a loose rein my horse will stretch and carry himself SO MUCH BETTER b/c that exercise teaches a horse to use his body and not be afraid of contact on the bit. He rarely lifts his head real high anymore and if he does I go right back to combing the reins with drag to tell him "Hey, wrong door."
One thing you need to be ready for is the fact that you may lose your back up when doing this exercise. But this is okay! Pick your battles wisely. Once your horse is consistantly relaxed and soft with that exercise then you can move onto shaping his body and asking him to carry himself in an actual frame. And actually you shouldn't have to use your reins to back up
He should go off your seat.
I'm going to disagree with driving him forward when he puts his head up. I dont' believe it's a stunt. I believe it's an unconfidence issue. If anything that might cause him to become more emotional.
One other improtant factor.........your riding. Make sure you are fluid in your body. NO BRACE. Don't jam your heels down, don't hollow your back. Move everything. If you are braced in your body then he will be braced in his body. Be fluid, keep your weight back, not on his forehand.