Reviewing your thread, sound opinions and advice were given consistently by KSman and your doctor. I certainly see your point in wanting to consider testosterone replacement therapy while also seeing your doctor's reasoning in not doing so.
A disadvantage of starting hormone replacement therapy is that it strongly tends to become an irreversible decision, making a person dependent for the rest of his life.
If it were me I would like to see what happens when estradiol is reduced to low 20's pg/mL before starting hormone replacement therapy.
I'm not confident that this will make for much improvement, as your LH and FSH are quite good already, but there's a chance.
If that did not work, I would insist on self-administration of the prescribed testosterone and use the minimum necessary to get free T to midrange, while keeping estradiol in the low 20's. This might succeed in preserving normal LH and FSH production, making it a reversible decision if desired.
I have a couple of other thoughts on your symptoms.
One is that you might be suffering from unsynchronized circadian rhythms. What this means is that there's a circadian rhythm not only for your wakefulness, which tends to be set by light/dark pattern, but every organ in the body has its own circadian clock. Very, very preferably they're in sync with each other. In some instances, such as shift work, diabetes, or obesity, the organs are typically out of sync. This has adverse metabolic consequences; also as not-fully-based opinion, it can result in strange body temperature patterns even when thyroid is good. (It is a fact that body temperature patterns can be strange and low while thyroid is good; it's opinion that circadian rhythm of the organs is related to this.)
Actions to take on the chance that this is related would include the obvious of honoring the natural light/dark cycle and having consistent sleep times, and also avoiding eating late at night and especially in the middle of the night.
Another is that there is some similarity in your experiences with chronic fatigue syndrome, which I don't say that you have, or even that it is necessarily one specific thing rather than a constellation of symptoms which has measurable markers associated with it and some response patterns. An important issue with CFS, which may be related to your situation, is the gut microbiota, which can be varied according to diet. There have been studies with fecal transplantation which I expect you have no interest in, but the point being, changing the gut bacteria can change outcome. I would not put too much stock into commercially available products let alone yogurts, though one of the refrigerated health store products might help slightly. Rather I'd look towards increasing microbial diversity from intake of various foods which have bacteria which may survive to the gut and which are associated with the natural human condition, such as found on vegetables, including those growing down in the dirt, which haven't been washed too much; and with home-fermented or home-style-fermented (commercial may be quite dfifferent) foods. There are also diets which tend to imbalance the gut microbiota, and ones which are restorative. Paleo type diets are restorative. High cyandin-3-glucoside intake also favorably affects gut microbiota balance.
The CFS-like hypothesis, while not necessarily right, would also tie in with your thyroid experiences.
At any rate those steps could not hurt and might help.