That’s happened to me a couple of times, and honestly, I haven’t had to do anything to convince them to put me back on them. I simply can’t function on even the most basic day-to-day level without meds (for me, Adderall — I’ve tried Ritalin, Vyvanse, Strattera, Provigil, Wellbutrin, Abilify, Clonidine, and at least one other that ADDitude lists as an ADHD drug. Adderall has been the only one that’s make a worthwhile difference, and it has to be the immediate release, not the XR). The combination of that being evident to my psychiatrist and having another doctor of mine back me up on the risk vs reward was enough that she didn’t even hesitate starting me back up.
Have you tried counteracting the spike in blood pressure by exercising more, eating better (check out the DASH diet for hypertension), and, if you smoke cutting down?