I recognize, talking to your doctor to be a difficult one.
If I had to do that I would start with a question: "Doctor X., if hypothetically a patient of yours believes in the benefits of parallel treatments, and wants to try them would you be willing to follow and work with them, in order to make their journey safer?" Then, I'll be hoping for an answer "Yes. Is that you?". Then comes the elaborate, but briefly presented (doctors have no time usually)alternative strategy,
where the Dr's role is emphasized as a key figure to keep general condition and lab results under good control.
This, I hope, would spare conflict, scare or withdrawal, as many Dr. could react that way. Some are surprisingly open and supportive. You probably know your doctor, already.
Also, I would rather let myself finish a course of fasting, and then go, in order to not deal with an explosion of denial, judgement or other negative opinion during my fast. All of these can challenge the process. A high/low TSH would not be that dangerous as far as you are functioning decently and your heart rate, blood pressure, mental process, are not drastically different from "normal" fasting lowering of BP, or fatigue. But, I am not an endocrinologist to know the exact numbers. All I know is that people have gone months and years without proper T3 T4 supplementation, or Thyroid suppression, whichever is the case, and are still living. You would not want to do that for months in a row, though. I would make a guess that a 21-30 day fast would be safe in any case, if you feel having decent body/mental function.
Hope that helps! Let me know how you dealt with your doctor.