I suggest giving the doctor a copy of my paper and then ask tp reduce your T4 med down to 100 and as a start, add 25 mcg of T3 in two steps about 2 weeks apart to give some time for the old T4 dose to dissipate. before the T3 kicks in fully. Then after 4-6 weeks see how you are feeling and add more T3 if needed, in increments of 10 mcg. TS3 is generic liothyronine. There is a brand name Cytomel.
Don't forget about the Vitamin D, B12 and ferritin. Very important.
Absorption of D3 is dramatically helped by takin magnesium! So you may try supplement with magnesium along with D3 and see if that helps your D3 levels. Also vitamin K2 helps in addition to magnesium.
Some important things to understand. TSH is used as a surrogate for FT4, however, it has only a weak correlation with FT4 (or FT3). So when TSH is suspiciously above range, a test for FT4 is used to confirm status. TSH actually has a negligible correlation with hypothyroid symptoms which is the reason patients go to the doctor. So, TSH is useful as a diagnostic only to identify overt primary hypothyroidism (damaged or diseased thyroid gland).
When taking thyroid medication, it spikes your levels to a peak about 3-4 hours afterward. That causes a suppressive effect on TSH that lasts for most of the day. A blood draw soon after taking the med will show a false low TSH and cause the doctor to possibly unnecessarily reduce med dosage. How could a suppressed TSH alone indicate over-medication if both FT4 and FT3 are in range? TSH has only a weak correlation with thyroid hormones in the unmedicated state, and worse when being medicated. So how then is a TSH supposed to indicate a patient's thyroid status? It doesn't. Forget TSH.
Doctors assume that all hypothyroidism is due to a damaged or diseased thyroid gland shown by a high TSH. This overlooks the majority of hypothyroid patients because hypothyroidism is best defined as "inadequate T3 effect in tissue throughout the body". It is the level of FT3 that determines your thyroid status. T4 is a prohormone, important only because it is available for conversion to T3. It is FT3 that creates the metabolic effect throughout your body.
For any patient diagnosed as hypothyroid, the usual treatment is T4 as needed to return TSH within range. This does not work well for most patients because the dosage is adjusted based on TSH, not based on symptom relief. Many patients taking only T4 med find that for several possible reasons, their body does not adequately convert the T4 to T3. That is what is happening to you. Your FT4 is at the top of its range yet your FT3 is only 24% of its range. A further increase in T4 dosage would be counterproductive. You don't want FT4 that high in the range. You would be much better off dropping your T4 dosage and adding some T3 med to raise your FT3 adequate to relieve your hypothyroid symptoms.
Along with that you need Vitamin D at least 50 ng/ml. B12 in the upper part of its range, and ferritin at least 100. You can confirm all this by reading my paper in the following link. It provides all the supporting scientific evidence needed. You might consider giving your doctor a copy and asking to be treated clinically, which means adjusting FT4 and fT3 as needed to relieve hypothyroid symptoms.
Before further discussion, what thyroid med and dosage are you taking?
Since you don't seem to absorb D3, would your doctor consider Injections as needed to get your D up to 50, where it should be?
Have you been tested for B12 and ferritin?