There is much to discuss but first please double check the range shown for Total T3. I have never seen one that is that tight. Usually it is more like 87-190.
As I mentioned I have never seen such a tight range, which makes me wonder about the applicability. Regardless you should insist on being tested for Free T3, along with Free T4 every time you go for tests. Free T3 is the biologically active thyroid hormone metabolized by all the cells of the body to produce energy.
Fatigue and weight gain are tow of the most common symptoms that can be related to hypothyroidism. Your Free T4 is only at 16 % of a range that is far too broad to be functional for everyone. If your Free T3 is similar in range to your Total T3, that is indicative that your body is trying to maintain thyroid function as best possible by converting extra T4 to T3. Many of us have found that we needed Free T4 at least at mid-range, and Free T3 in the upper third of its range, and adjusted from there as needed to relieve hypothyroid symptoms. Having a low TSH along with the Free T4 makes me wonder about the possibility of 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 to produce thyroid hormone. Inadequate pituitary function could also correlate with having low prolactin, since prolactin is produced by the pituitary gland in the brain.
A good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free t3 levels as needed to relieve symptoms. Symptom relief should be all important, not just test results.
Hypothyroid patients are frequently deficient in Vitamin D, B12 and ferritin. You need to supplement as needed to get your D up to at least 50 ng/mL. B12 is especially important to prevent fatigue. B12 should be in the upper end of its range. Ferritin is also very important and it should be at least 100. If not tested for those you need to also get those done and supplement as needed to optimize.
If your doctor is unwilling to do these tests and treat clinically then you will need to find one that will do so.
You could give a copy of the entire paper in the following link and ask the doctor to re-consider his diagnosis and treat you clinically, by giving you a trial of thyroid med adequate to raise your Free T4 and Free T3 levels into the upper half of their ranges. From what you have said about that doctor, however, I have serious doubts that he would consider the possibility that he is wrong on anything. Endocrinologists tend to be very rigid in their "Immaculate TSH Belief", by which they only pay much attention to TSH. That is very wrong. If they go beyond TSH it is usually only to test for Free T4, and then use "Reference Range Endocrinology", by which they will tell you that a thyroid test result that falls anywhere within the range is adequate. That is also wrong. In addition, I think that the larger the group of Endos, the more likely it is that none will risk straying from the group's standard of care for hypothyroidism.
I highly recommend reading at lest the first two pages of the link, and more, if you want to get into the discussion and scientific evidence for all that is recommended.
If you find that you do need a good thyroid doctor that will treat clinically, as described, please give us your location, and perhaps we can suggest a doctor that has been recommended by other thyroid patients.
I'm not sure about that fatigue symptoms with vitamin D deficiency since I always seem to have other health issues going concurrently causing fatigue but my notable symptoms of vitamin D deficiency include bone pain notably in my fingers, poor immune system so I catch everything going around (when I was down to 30 nmol/L (12 ng/ml) I was sick basically non stop for 6 months with cold, flu, strep throat twice, bronchitits), mucus build up in my throat (constantly having to clear my throat), calcium deficiency (which in itself causing a lot of symptoms), muscle weakness in the legs.
I do have mood issues and irritability and some mild depression but put that down to other reasons. I hadn't realised it could be a possible symptom of low vitamin D levels. Hmmm interesting. I find it not so easy to lose weight but I have been sneaking sugar lol.
I found an article on the ten most common symptoms of vitamin D deficiency...
University Health News - 10 Vitamin D Deficiency Symptoms You Can Identify Yourself...
"Here’s a list of 10 of the most common Vitamin D deficiency symptoms:
Bone softening (low bone density) or fractures
Fatigue and generalized weakness
Muscle cramps and weakness
Joint pain (most noticeable in the back and knees)
Blood sugar issues
Low calcium levels in the blood
Mood changes and irritability
I forgot to add I have had issues getting my vitamin D levels up. The first time was due to serious magnesium deficiency (needed for vitamin D absorption). And taking high doses of vitamin D worsened my existing magnesium deficiency. I currently have this issue again being low in vitamin D despite taking 3000 IU daily. The reason this time is due to stress lowering my stomach acid which is affecting absorption.