"Will it help if I take T3 while dieting to compensate for the metabolic adaptation?"
Consider this:
“Protein conservation during prolonged fast: a function of triiodothyronine levels”
"During the control fast, the mean serum T3 concentration fell to 61% of baseline, the mean serum RT3 concentration increased 100%, and there was no change in mean serum T4 concentration. These changes are similar to those described in several previous reports of fasting. Administration of small doses of T3 approximating the normal daily T3 production rate resulted in maintenance of serum T3 concentrations slightly above the baseline levels, but within the range of normal, for the duration of the T3 fast.
In the present study, preventing the fall in serum T3 concentrations significantly increased urinary urea excretion as compared with a control fast, suggesting that the fall in serum T3 concentration in fasting and illness is a protective mechanism limiting muscle protein catabolism. This suggestion is supported by a study, recently reported in abstract form, in which the administration of slightly higher doses of T3 during a prolonged fast resulted in increased muscle glutamine and 3-methylhistidine release and increased urinary urea and ammonia excretion."
I would not recommend T3 for those not taking AAS. The risk of muscle loss is too high. I also know lifters not on AAS who have dieted with and without T3. With (reasonable doses of) T3, they lost more muscle (when BW went down, there was no corresponding improvement in how they looked compared to when they dieted without T3).
There is also more to metabolism than just T3 and leptin.
“Hypermetabolic low triiodothyronine syndrome of burn injury”
"T3 treatment restored FT3I but did not affect resting metabolic rate (MR), compared with placebo therapy. The hypermatbolic response to burn injury appeared to be independent of thyroid hormones."
This brings me to my last point with regards to those who have dismissed the potential benefits of refeeds (on metabolism), and justifying that based on studies which show refeeds don't sustainably/significantly improve surrogate markers like T3, leptin, and various other hormones. These are surrogate markers and probably don't capture what is happening with the metabolism. Given that (as long as food, fluid, sodium, sleep are consistent; and weighing first thing in the morning after urination) BW loss is pretty highly correlated with fat loss on a good resistance training program and diet, I would argue a better way to assess whether refeeds help the metabolism is by tracking how BW changes day to day.
It's not unusual that if refeeds are implemented effectively—and while it may run counter to what you would expect due to the increase in weekly calorie intake compared to not incoporating refeeds—they actually not only do not hinder fat loss, but they help it.
My practical takeaways are not to worry about too much about thyroid hormone when dieting. While there are other factors that contribute to metabolism, thyroid levels will decrease markedly during a diet (metabolic adaptation). However, they recovery swiftly once calories are increased after a diet.
Many hospitalized ill patients (especially those in an intensive care unit [ICU]) have low serum concentrations of total thyroxine (T4) and triiodothyronine (T3), and their serum thyroid-stimulating hormone (TSH) concentrations are typically low, but may be low-normal or normal. Almost all patients who have a subnormal but detectable serum TSH concentration (greater than 0.05 mU/L and less than 0.3 mU/L) will be euthyroid when reassessed after recovery from their illness. Some hospitalized patients have transient elevations in serum TSH concentrations (up to 20 mU/L) during recovery from severe nonthyroidal illness (indicative of recovery).
Even many bodybuilders who take T3 during contest prep and immediately discontinue T3 after the prep experience swift recovery of the endogenous hypothalamic-pituitary-thyroid axis—within days. This is in stark contrast to AAS use where endogenous testosterone production takes a while to recover after discontinuation of AAS (which is the reason many enhanced bodybuilders recommend a PCT if coming off gear).