Your problem is obvious, your doctor just has no idea what he’s doing and is very common. When SHBG is high, Free T isn’t and a Total T of 540 with an SHBG of 65 just will not do. You need to get Free T to high normal and you will likely need levels in the supraphysiological ranges to ever have enough Free T, sadly you may need to go private as androgen therapy is seriously lacking under most insurance networks to do a lack of doctors specializing in TRT.
A doctor that doesn’t test estrogen is a sign you are working with an incompetent doctor. Defy Medical, a telemedicine anti-aging clinic is a popular option.
What about thyroid testing TSH, Free T4, Free T3, Reverse T3 and antibodies? Thyroid problems share the same symptoms as low testosterone, TRT requires a thyroid that is optimal as the two work together to restore metabolic rates.
If your doctor says thyroid is normal or in range, in range or normal isn’t good enough.
Google “Optimal VS Normal Thyroid Levels for all Lab Tests & Ages”