Obtaining gait is a complex problem because there are many things that can prevent it, including poor breeding.
First, review the basics (correctly fitting and adjusted tack, anatomically correct hoof trimming, normal shoeing, horse fit and with no pain isssues, etc.). None of these things will make a horse gait correctly, but any one can prevent it.
Second, review rider practices. Ensure the rider is sitting squarely and balanced (a la Sally Swift).
Most Walkers are laterally gaited. They tend to the pacing side of the gait continuum. This requires that the back is neutral to slightly ventroflexed. If you bascule a horse like this they will trot. They can't help it; it's equine biomechanics. So when we talk of "collection" in laterally gaited horses we don't mean "bascule" we mean "not strung out." Folks moving from trotting horses sometimes need some time to learn to feel and understand this difference.
You are correct in pushing the horse to the bit with seat and leg and "feeling" the mouth. This is a good beginning. Many gaited horses like to work off the bit and don't do well on loose reins (some other do quite well that way).
(I've not asked about the type of bit. Personally, I don't like the 8" shank monsters that many Walker folks use. Nor do I care much for twisted wire mouthpieces. This is, however, a separate discussion.
If you keep the hand soft you can use what you like. Of course what you use will have an influence on how the horse goes, how it turns, etc. The bit should be appropriate to the task at hand. Just more of the complexity, don't you know.
If dropping the rein causes a negative change in way of going then don't drop the rein (I don't care what an earlier owner said or what some trainer says or what some self styled expert says). If you drop the rein and the horse changes in a way that is not satisfactory then don't do that.
You ride a Walker just like you ride any other good horse. You sit in the middle, stay balanced, and keep the aids (hand, seat, leg, balance) in accordance.
You might want to have someone video you on the horse. The video camera is the greatest rider training device ever invented. It's also the most humbling, because the lens does not lie.
Good luck in your project.
P.S. If you don't have Lee Ziegler's book then get it.