Strength and balance, you MUST have.
But more important, is the giant brace in the neck, the ribs and the loin that your description positively screams:
stiff-legged, head up (and the upside-down necks to prove that it’s a long-standing thing), 4-beat, no suspension (none), forehand thing.
You’re going to have to figure out how to get the horse(s) to turn loose of these braces. I can almost tell the horses balance when they canter is not just on the forehand, but over the inside front leg as well (classic ‘dropped shoulder’ feeling).
So…lots of lateral bend at the poll on a very tiny scale: ask for the eye, making sure the ears stay plumb and the horse turns loose from the rein, not just swings it’s head sideways, or not pulling the rein sideways, not staying on a contact but releasing when the horse gives.
Ask for the HQ to give laterally. This is difficult to get right. Just about anyone can get a horse to ‘disengage’ but it takes some chops to get a horse to untrack behind, keep his walk rhythm, unweight his inside legs (and therefore weight his outside legs), and turn loose in his hips and ribcage. The purpose of ‘untracking’ is not to disengage the horse, but rather to engage him, to lighten that inside front leg and take weight on the outside hind (your canter strike-off leg), to get him to fill your outside rein so you can ‘power up’ forward into a balanced canter.
(Of course, you don’t untrack the HQ only for a canter transition. Rather, the lateral rebalancing gets the horse into your outside rein, it can set him up, engage him so you CAN go right to canter, or go to a lively rollback over the haunches or any other athletic move you need.)
I suspect this won’t get very ‘right’ until you can get the HQ thing just right, and there are soooo many ways that horses fool you into thinking it’s right when it is still braced up physically and mentally, fleeing pressure rather than rebalancing and operating on a feel. Having the inside rear leg cross over the outside, behind…well, you do need that but even if it’s happening, it can still be done without the horse releasing his brace.
That’s where I’d focus, and you’re probably going to need some excellent help to get it working right. (Your local mentor maybe, definitely mine on the other side of the hill- who will drive his car over the pass so you don’t have to trailer horses two hours one way).
But once you DO know how to get it working (it can take a while!), you’re gonna have some CHOPS, honey.
Once the untracking is good, bring on the lateral work- leg yield, shoulder in, expanding circles.
And I’d continue to hand-gallop outside given the opportunity.
And then, you will be able to develop the strength these downhill stock types need to lift their forehands and canter beautifully. They can.
But the part where your back is getting pounded…you’re asking the horses to ‘go anyway’ while there’s a big giant brace in them. It isn’t any more pleasant for the horse than for you, I don’t think!