I have the same problem with my hands, but it is not because I am using them for balance. It is because I am not letting them be springy enough at the elbows. They float up when I rise, they come down when I come, because I'm not changing the shape of my arm.
If this is what you are doing (and I am green, so I can't tell from the vid - just from the conversation) then maybe what I do will work for you too. We decided it was better for me to keep my hands a little too low all the time at a trot, just so I can get used to what my elbow and shoulders need to be doing. So I make sure that when I'm trotting, I can always feel the hair of his mane on the backs of my knuckles as I'm posting. Not all of my knuckles - :) - just the bottom ones that would sensibly be touching the horse if I were holding the reins properly with my hands a little low.
So I let the rest of my body do a proper post, but as long as I can feel hair, my arms are going to be doing more or less what they are supposed to. It is not easy, but it is helping!