I take low dose Armour Thyroid every morning prescribed by my Endo for Hashimotos. My TSH without any med is right on the border (3.8 to 4.0) and with Armour is consistently at 1.0. I just have high levels of both Hashis antibodies which we are trying to lower. My Endo told me (back in Oct 2013 when it was first diagnosed and prescribed) that I cannot take any "minerals" for four hours after taking it or it can decrease the absorption of the med. If my memory is correct, he mentioned Calcium, Iron, and Magnesium.
Have you had any inflammatory cytokine panels done? The reason I ask is because if you can lower these with immune modulators (assuming you can't find the root cause), then hypothetically you can quite the autoimmune response.