While you say the vet checked him over? How thoroughly did he/she do that? Were X-rays taken of his spine/hips/withers process? Was blood pulled? Was his jaw X-rayed to make sure bars aren't broken, and have his teeth been done, with X-rays taken of the mouth/teeth too? How long has it been since his sheath was cleaned?
At this age, 19 is not young. It would be a rare horse that doesn't have some arthritis issues, and for all you know, there could be broken ribs, or vertebrae too. Without X-rays you are flying blind basically, wanting to blame this on training rather than looking at the age of this horse, and the body issues he may have.
Other things could also be peritonitis, your girth is too high and compromising blood vessels, your saddle may not fit with your weight in it, you could be toeing him when you mount, or dragging toe across rump, or coming down too heavy in saddle.
Chiro/massage might help, so might acupuncture, especially that done with electrical stimulation, but without more thorough vet exam, X-rays and such, you have no clue what is really going on with this horse.
Find a good horse vet, one that deals only in equines, or take horse to vet school and get a complete workup done. There are so many things that could be wrong, and without knowing for sure there aren't broken bones, major infection issues, or abscesses deep within muscle tissue, to only focus on training as the problem is a disservice to the horse.
Horses make me a better person.