Didn't your doctor comment on your low FT4 level and your TSH level being associated with hypo symptoms such as those you mentioned?
With a suspicion of Hashi's, the tests to be run are for the thyroid antibodies, TPO ab and TG ab. I also strongly suggest that you get tested for FT3, as well. If the doctor resists then insist on it and don't take no for an answer. FT3 is the biologically active thyroid hormone that largely regulates metabolism and many other body functions. Scientific studies have also shown that FT3 correlated best with hypo symptoms, while FT4 and TSH did not correlate very well at all.
Be aware that just because your appointment is with an Endo, does not mean that he will be a good thyroid doctor. Many of them specialize in diabetes, not thyroid. Many also have the "Immaculate TSH Belief" and only want to diagnose by TSH. This doesn't work. A good thyroid doctor will treat a patient clinically, by testing and adjusting FT3 and FT4 as necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not thyroid test results. You may want to inquire about the doctor's willingness to treat you clinically in this manner so that you will know if you need to keep looking for a good thyroid doctor.