Interesting you say through walk. I am not by any means arguing that it should be through walk. I have been taught to do it through walk and do it through trot.
The reason I said through trot at this point is it sounds like she actually has the changes, but still only gets them with a pole. I was taught when you get to that point where you want to take away the pole to think of dropping to a stride of trot then back to canter and once you have that you can just get the big half halt to signal for the change.
But I do see where you are coming from in taking it back to the walk. Which actually does make sense.