Well you have autoimmune hypotiroidism. But you already know that.
It is strange that your symtoms did not improved even though you are on levotyroxine. The logical step for your doctors is considering another disease hidden behind the hypotirodism.
So definitively, do not start the trial until you talk with your doctors, wait until your endocrinologist considers it appropiate. I belive that more tests are needed. Ask your doctors EVERY QUESTION YOU HAVE. Start making a list right now and take it to your appointments tomorrow.
If you are not confident, then look for a second opinion. Gather all your results an go with another specialist after tomorrows appointments. This diseases are very clinical and personal evaluation is of upmost importance.
I recommend another endocrinologist, or if you are afraid of a biased opinion than an integrative type especialist would do the trick. Internal medicine if you are less than 65, geriatrist if you are more than 65.