Hm, well, I wouldn't say "cheating" but a crutch. . . like you I don't like anything you have to keep doing over and over, doesn't seem right.
Nevertheless, it's wonderful that you have found a gentle way to communicate what you want! Now you can ask him to go on without the squeeze. As soon as he tenses up, or pops his head, stop, and apply the squeeze again, as if to say, "Remember this?" Practice a few strides, then proceed without the aid.
I'm doing something similar to this right now. By horse is a "head popper" too, even in so simple a thing as going from walk to trot. I can walk her in a very relaxed state, then ask for trot, and the moment she tenses, I release my aids, tell her to keep walking, "We're not ready." It takes awhile sometimes before she'll do transitions fairly relaxed, without tensing.
By the way, we do warm up some before focusing on the "stay relaxed" exercises. She wouldn't possible stay relaxed for the first trot of the day. It's a habit, and it's going to take some time to get over it.