Well, maybe the neck inflammation is also causing your low BP, perhaps via histamine that is released from basophils as part of some other associated autoimmunity. And
when BP goes low, your body increases heart
rate to compensate - to preserve the rate/pressure constant.
(You are probably aware of the association between chronic urticaria and Hashimoto's, e.g.)
What if you tried an antihistamine like diphenhydramine, just to see what happens to your BP and HR?