Yes, I'm familiar with all this.
Your doctor is one of the few who looks at T3 levels and put you on Liothyronine, which is T3 and does not need conversion.
However, like most conventional doctors, does nothing to address the underlying cause (in your case the conversion of T4 to T3 problem).
A fast 101 on T4 to T3 conversion.
Necessary for the conversion are:
---The enzyme deiodinase, the minerals zinc, selenium and the anti-oxidant glutathione are essential . Any deficiencies in these might impair T4 to T3 conversion and consequently thyroid function.
Furthermore, Magnesium, NADPH, Thiamine, B6 in P5P form, B12 as Methylocobalamin, vitamin C, vitamin E , cysteine, glutamine and glycine,
play a synergistic/supportive role to varying degrees.
Again any deficiencies in some of the above will impact T4-t3 conversion.
---Digestive and/or G/I issues will affect the absorption of many of the above nutrients and vitamins.
---Liver function must be optimized, as some of the conversion and some thyroid associated enzymes take place in the liver.
Again, beware of conventional doctors may test you and find no issues with the liver.
The blood test results simply may only indicate absence of serious liver disease and not diminished capacity to convert T4 to T3 and/or thyroid--associated enzymes.
---Finally, should you have high stress levels or other factors affecting your cortisol and adrenals in general, this would contribute to thyroid conversion problems and
reduced thyroid function.
Free T3, Free T4 and Reverse T3 (RT3) tests will accurately indicate thyroid function and the RT3 will indicate possible adrenal issues.
If you need more details or have any questions, please post again.
I hope this helps, however, please note that my comments are not intended as a replacement for medical advice.