However, the extreme of temperature swings I am experiencing seem to always correspond with M.E. flare ups.
You're not alone in that. In the early stages of my ME, when I was having ME flare-ups several times a day, my body (at least oral) temperature would abruptly rise at least few tenths, and occasionally up to a full degree C. I usually felt feverish at those times. Sometimes I'd feel feverish but find that my oral temperature was below normal, which is what makes me think that our perceptions involve temperature differentials.
I think we normally perceive only reasonable significant body temperature changes. Since ME seems to mess up neural functions, it seems plausible that the perception neurons are overreacting.
You could search online to see if there are any TCM, ayurvedic, or other remedies listed for exaggerated hot/cold sensations. Surely other people throughout history have suffered that symptom before, and maybe someone noticed that a certain herb, or tea or whatever helped. Those other medical systems may have been better at noticing accidental treatments and recording them. Conventional medicine is too focused on theories, as well as on finding applications for patented drugs, and on profitable markets. Funding for diseases probably correlates strongly with potential market size.
Another suggestions: experiment with preventive treatments. Instead of trying to treat the feelings after they've occurred, try things before you expect to get a flare-up. If you expect to get chilled while fetching the mail, try soaking your feet in hot water first, or taking a small amount of hot spice. If you expect to overheat while cooking, maybe a cold pack on your head before starting would help. I admit that the chances of discovering something useful are low, it's better than no chance at all. The few things I've discovered in my 19 years of ME have certainly improved my quality of life, so I've very glad to have found them.
While it's not helpful to us as patients, I do understand how frustrating it must be for doctors not to have answers and finding themselves unable to offer treatments or solutions to all of the various symptoms we pose.
That's the feeling I've gotten from several doctors too. Unfortunately, some turn that into anger at the patient.