In agreement with anebel. Try to get him moving forwards more to counteract the upwards movement by encouraging him to stretch his neck long and low and by doing lots of quick transitions to get him using his quarters more, but be careful not to push him too much otherwise he'll fall onto his forehand. Little and often should do the trick because you're asking him to use muscles he obviously hasn't been using or using properly.
My mare also has a bouncy trot and canter so when I rise to the trot I have to ensure I rise, then push my pelvis forwards before coming back down otherwise I could easily 'bump' back into the saddle. Trying to sit and ride 'lightly' also helps and allows her to use her quarters and when warming up I spend a bit of time in the forward seat in both trot and canter to allow her to move freely and loosen up her back and shoulders.
Good luck with Zahir. He sounds lovely and what a great project to work on!