Well all that is a mighty tall order for a medication to fill. Generally, most everything has sideffects and I think we all risk some kind of internal organ compromise when we decide to take anything at all. Either medications or anything in the herbal world too.
I suppose it comes down to deciding how much damage the medication will do as opposed to unaddressed relentless pain, which can have it's own devestating effects on the body.
For period pain you need to see a good gynecologist who had plenty of experience with the problems of Endometriosis and Adhesion formation or better yet, a Reproductive Endocrinologist.
It would all depend on the exact cause of your pain as to what might be your answer.
I have HBP too. Mine got really bad when my female situation hit a crisis point.
Pain it self will make your BP go nuts and if you are not getting proper pain control it might not be the substance but the lack of good pain control that is messing with your BP. Also, if it's a hormonal problem there goes the BP again. Period pain is extremely complex and really needs to be addressed through a specialty medical person.
I get brain chemical meds. A SSRI/Tricyclic antidepressant combo, anti spasmodics that work on specific internal organs, anxiety med, pain med, and an anti diabetic med which amazingly contributes to me having less pain because my blood sugar is not dropping suddenly sending my body into spasm.
And, since I've brought a cardiologist on to tend my health, the Beta Blocker he put me on has cut the intensity of my migraines in half.
I get a specific migraine med for when I get to the vomit stage of the migraine but since the beta blocker came on board my episodes are further between.
So for me the beta blocker handles more than one problem and it keeps my heart out of Tachycardia.
The anxiety and pain med I get are actually taken on and off only on the days when my problems override the daily meds I have been given for body support.
I also take those less often than in the past.
I found the best help for my female pain problems with the Reproductive Endocrinologist, the Gastroenterologist and the Cardiologist.