Ok, so wrapping the throat latch around the cheek pieces turns out to be surprisingly effective (more than just running it through the brow band).
On normal horses where it doesn’t rotate, just putting the throat latch over the hackamore headstall works great, but for the one where it always rotates a ton I have to actually wrap the throat latch around it once on each side and make sure it’s wrapped snuggly so the stiffness of the throat latch holds it in place. Also, I kind of wrap downwards so the throat latch sits further down around the jaw, which also helps. Your throat latch has to be a bit long to start with to avoid ending up too tight after you wrap it (if you just put it over top it doesn’t need much extra length, maybe one hole extra).
I was also able to get the bailing twine option to work by sliding it further down the cheek pieces and making it snug to the point where it was pulling the cheek pieces down a bit. (Did not bother Miss Picky to have it snug and she definitely would have told me.) so if you don’t have extra throat latch length then that’s also an option.
I haven’t tried this yet but I think you could create some kind of loop on the cheek pieces that you could run the throat latch through to get the same effects as wrapping without needing extra length (and really simple to do, instead of getting the wrap just right). I’m just considering how to add the loops so they will be fixed in place. (Current thoughts involve the leather punch.)