Could it be an issue of protecting the horses from being overworked??
I know 2'6 may not seem much, but even if (hypothetically) the horse is doing just 2 lessons per day every second day at that height, it is still a big ask for him.
They have to do what is best for the horses. It may be holding you back, but if they were to let you progress, then they would have to allow the other students as well. They have to draw a line somewhere for the well being of the horses.