I'm sorry to admit to you that your post made me laugh out loud. You described PERFECTLY my own experiences with the blasted trot. I know you are frustrated and discouraged, but your description was hilarious. Again, sorry.
Frustrated and discouraged? Been there!! Slowly improving...actually things are quite a bit better now. Some things to think about :
How is this horse's comformation? Ie, is he dippy backed and,or downhill from croup to withers? This kind of conformation , which my horse is,,,throws the rider into an incorrect position which causes your balance to be off and makes a sitting trot quite a challenge. It tilts the front of the saddle down-ish. If you can shim and get the saddle more level , it helps a lot.
Some horses go more 'hollow', ie don't bring their backs up, push the nose outward..and this makes the sitting trot harder
The person's natural response to these factors , imho, is to lean slightly forward at your shoulders and lift your hands, which directly contributes to the bouncing out of the saddle , even your feet leaving the stirrups as you bounce uncontrollably. It's terrifying ! It's a vicous cycle of trying the wrong things to get your balance and control back.
So, what's helped me : get the saddle in the best position possible
keep your shoulders back
keep those heels down as you already mentioned as your baseline position, but **see pedaling below***
relax (singing, breathing)
'pedal' with your heels in time with the movement of his back feet,,up/down/up/down. Ideally as his right back foot lifts to move forward, you lift your left heel up. Your pedaling is not pushing the heel down (it should be 'baseline' down, but is raising it up, then returning to the baseline 'down' position) It might be necessary to have her call out when his right foot lifts, so you know exactly how to get in rhythm with him. When you get it right, this technique can help you sit the roughest trots. It causes your body to move more with his movements, if that makes sense. Don't push down hard enough with your heel to pull your hip way over/down to one side, just a firm but gentle pedalling is the ticket. I learned this mistake first hand trotting in my bareback pad, nearly made myself fall off ,,,but at least I wasnt bouncing....
more trotting on the problem horse. I once made sure I stuck with it for at least 20 mins of trotting and his trot got softer by the time we were done. The horse will seek harmony with the rider, if possible, I mean , it's not very comfy for them either, when we're banging down on their backs. After that, I got a better trot , more often from him. That, plus the other techniques have helped us a lot. He rarely seems to be in that bone jarring trot these days,,,or else my muscle memory, and ???whatever has occured,,,has helped.
If we ask for , or let them, stop trotting after a short stretch (natural when you don't feel safe and it's painful!!), then they're not motivated to do their part to make it better. Imho.
Have you tried slowing him down, but maintaining a trot. That will make it less jarring, probably.
I feel your pain! I almost decided I just couldnt do a sitting trot on Sonny. I nearly resigned myself to always do a posting trot. But, I'm stubborn....and got help from and instructor...
Hope something I said might be helpful...
Good luck!! Hang in there!! Keep us updated on your progress