Do not hesitate to ask about pituitary function. Your symptoms along with diminishing TSH and the low FT4 seems likely to be central hypothyroidism. With central there is a dysfunction in the hypothalamus/pituitary system resulting in TSH levels that are too low to adequately stimulate the thyroid gland.
Besides pituitary function testing you need to get some additional tests done. I recommend that you should always test for both Free T4 and Free T3 each time. In addition, hypothyroid patients are frequently deficient in Vitamin D, B12 and ferritin, so you should get those done and then supplement as needed to optimize. D should be at least 50 ng/mL, B12 in the upper part of its range and ferritin should be at least 100. Also, due to the potential pituitary issue, you should be tested for cortisol.
When you have additional test results, please post here along with reference ranges and we will be happy to help interpret and advise further.
I have no experience or knowledge gained from others' to say about a diminishing TSH with central hypothyroidism. I can say that you have a number of deficiencies in your test results. Just because a test result is within the so-called "normal" range does not mean it is adequate for you. First because the ranges are far too broad due to the erroneous assumptions used to establish them. If you want to confirm this and understand why the ranges are not functional for many people, have a look at Item no. 3 on page 4 of the following link.
In trying to assess a person's thyroid status, TSH is an adequate indicator only when at extreme levels. The main value for TSH is usually just to distinguish primary from central hypothyroidism. the most important consideration is symptoms and you have mentioned several that are frequently related to hypothyroidism. When typical hypo symptoms are present, and Free T4 and Free T3 are in the lower part of their ranges, then hypothyroidism is likely. Your FT4 was only at 30% of its range, and your Free T3 only at 3 % of its range.
Since the correct definition for hypothyroidism is "insufficient tissue T3 effect due to inadequate supply of, or response to, thyroid hormone", we also have to look at variables that are major contributors to the response to thyroid hormone. that is why it is important to look at cortisol, Vitamin D, B12 and ferritin. Cortisol should be neither high nor low. Vitamin D needs to be at least 50 ng/mL, B12 in the upper end of its range and ferritin should be at least 100. Your D should be okay with supplementing as you do daily. We'll know more about B12 from the test result. I can't understand how you could be taking 150 mg of iron daily and still be deficient in ferritin. Are you sure that is 150 mg or mcg?
By the way, hypothyroidism has been shown to reduce cerebral blood flow to 4 key areas of the brain, including the area affecting cognitive ability. There was also a study that concluded that cognitive impairment from hypothyroidism was reversible.
The most important thing you need is a good thyroid doctor. By that I mean one that will diagnose and treat based on symptoms, instead of just based on TSH and FT4 compared to the flawed reference ranges. So you are going to have to find out if your doctor is willing to do that. If not, then you will need to find one that will do so.
Your doctor is just delaying action that you need. Waiting is not going to help clarify anything. It is already clear that you need to be on thyroid medication adequate to raise your FT4 and FT3. Note the following conclusion from an outstanding thyroid doctor. "The free T3 is not as helpful in untreated persons as the free T4 because in the light of a rather low FT4 the body will convert more T4 to T3 to maintain thyroid effect as well as is possible. So the person with a rather low FT4 and high-in-range FT3 may still be hypothyroid. However, if the FT4 is below 1.3 and the FT3 is also rather low, say below 3.4 (range 2 to 4.4 at LabCorp) then its likely that hypothyroidism is the cause of a person's symptoms." Of course the reference ranges and results are in different measurements than yours, but the inference is clear.
Does it really make any sense to say that a FT4 or FT3 test that is at the very bottom of the range (like yours) is equally acceptable as one that is 100% higher, at the very top of the range? No it does not. The ranges are not based on results from adults with no thyroid pathology. The ranges are based on all test results at a given lab, with the only results excluded being those with high TSH. In addition everyone has different thyroid parameters at which they function best. So you are going to have to change your doctor's mind by giving him relevant information and asking him to read and reevaluate his diagnosis and treatment protocols. A good start would be to click on my name and then scroll down to my Journal and red at least the one page overview of the full paper I linked for you previously. If that overview and the full paper does not change your doctor's approach, then you will need to find a good thyroid doctor that will diagnose and treat clinically, based on symptoms rather than by TSH and FT4 compared to demonstrably flawed reference ranges.
Also, if you cannot get your ferritin level up by the supplement you mentioned then the doctor might be willing to give you iron infusions. Ferritin needs to be at least 100.
Just being an Endocrinologist does not assure being a good thyroid doctor. Many Endos specialize in diabetes. Many also have the "Immaculate TSH Belief", by which they really only pay attention to TSH for both diagnosis and treatment. That does not work for most hypothyroid patients. To best prepare for the next step, I highly recommend that you read through the info I gave you above and take a copy with you to the doctor and ask him to read it and consider clinical diagnosis and treatment, based primarily on symptoms, rather than just TSH and Free T4.
Please let us know how things progress for you.
I had already responded to your other message.