Well those symptoms can be thyroid related. To get a better idea of what is going on he also needs additional testing. Before going further, I should clarify the correct definition for hypothyroidism. This is not just insufficient thyroid hormone, instead it is "insufficient T3 effect in tissue throughout the body due to insufficient supply of, or response to, thyroid hormone." Note that the response to thyroid hormone is also important. Also note that Tissue T3 Effect is what determines a person's thyroid status, not TSH and FT4. A person must have adequate T3 at the cellular level plus optimal Vitamin D and cortisol, among other things. Apparently your husband has not even been tested for Free T3. How about Vitamin D and cortisol?
T3 availability is mostly dependent on conversion of T4 to T3. Sometimes under the wrong conditions, T4 is converted to excess Reverse T3, instead of T3. Conversion is affected by a number of variables including ferritin and B12. So the bottom line is that your husband needs to be tested for Free T3, Reverse T3, cortisol, Vitamin D, B12 and ferritin. Free T4 and Free T3, the biologically active thyroid hormones should be tested every time he goes in for tests.
As for the question about the dose being doubled with little effect on serum TSH and FT4 levels, first I cannot believe the doctor would just double his prior dosage of 137 mcg all at once. He could have had a severe reaction. Fortunately T4 med has a half-life of about one week. which means that it takes about a week for serum levels to reflect even 25% of the increase. As to why even that did not show up in followup tests, might be related how he took the med, or poor absorption, or test variability, or excess conversion of T4 to Reverse T3, or other unknowns. Of much greater importance as I see it is that the doctor should not have been dosing him based on TSH, as I am sure happened since his FT4 was mid-range. Instead dosage should be based on achieving symptom relief. This usually requires Free T4 at mid-range, and Free T3 in the upper half of the range, and adjusted from there as needed to relieve hypo symptoms. Your husband's FT4 was adequate, so why double the T4 med? The doctor should have been doing the needed additional tests to better understand what was going on and then make a decision on med and dosage.
A good thyroid doctor will diagnose a person clinically, by first evaluating for symptoms that occur more frequently with hypothyroidism. Then he will do extended biochemical testing as required to confirm what the symptoms indicate. Then he will treat clinically by adjusting Free T4 and Free T3 levels as required to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important in treatment, not just test results, and especially not TSH results. Based on your experience with that doctor I seriously doubt he will accept these suggestions and change his protocols. If you think it might help you can click on my name and then scroll down to my Journal and read the one page overview of a full paper on Diagnosis and Treatment of Hypothyroidism: a Patient's Perspective, and consider giving a copy to the doctor. If you think it is unlikely that the doctor will do so, then let us know your location and perhaps we an suggest a good thyroid doctor in your area, that has been recommended by other thyroid patients.
First thing to understand is that when taking thyroid medication, TSH cannot be used to determine thyroid med dosage. Type of med and dosage should be based on relieving symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. A recent, excellent scientific study concluded that, "Hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and FT3 elevated further into the upper half of its reference range."
So before going further, please tell us about all the symptoms he has.