There is plenty of literature available to educate yourself on fasting to calm your fears.
Your doctor will never condone fasting. Would your mortgage broker help you with a program that allows you to instantly pay off your house? Wouldn't be in his best interest, would it.
It's a very rare doctor indeed who will be more concerned with the patients health than with the bottom line or his ego.
Taking thyroid meds does add a wildcard to the situation. You're on T4 only, so that isn't as dynamic. The half life is about 2 weeks. If you do a short fast (week or so), I would think that you can continue dosing as usual. Longer fasts may require a modification or two during the fast. Dr Fuhrman recommends cutting the meds in half:
I'm on Armour, and occasionally I have to drop my dose down due to heart palipitations. The T3 is very active and the half life is only a day. But I do continue to dose while fasting. Hopefully the fasting will eventually cure this thyroid mess.
For sure you should switch your meds to Armour. Why would you take synthetic when you can have natural? Armour doesn't only have T4, it has T3, T2, and T1. All are necessary and sometimes difficult to convert in thyroid patients.