The kickback you are seeing is derived from a horse’s natural propulsion and talent, ergo, that back snapping movement is simply something the horse possesses, or not.
You can do a million exercises to tidy that hind end up and make it more efficient, but you are going to have a tough time changing his style to Gem Twist.
Hill work will strengthen the hind end, particularly the stifles. Poles will engage the hind end and strengthen the topline through the back, the longissimus dorsi in particular seems to benefit from lots of pole work - eight trot poles in a row, etc. Setting up a course with low, wide facing jumps will teach the horse to break over himself and clean up the hind end.
I prefer grids, mostly, for this type of horse - something that I find works is to place a big jump (bigger than the grid) somewhere unrelated to the grid - canter over it and do a neat, soft rollback to the grid. The grid will be ascending in height, first a vertical, then a slightly higher vertical, then a slightly higher but much wider oxer, to a larger/wider swedish oxer, to a second swedish oxer that is the height of the unrelated jump.
Grid Legend:
- : 1 stride
–: 2 stride
| : vertical
||: oxer
{}: unrelated large fence
So the grid would look something like this:
v < – {} <----
v
v —> |-|-||–||–|| ------>
Alternatively, you can change the distances and make the first two jumps bounces.