I've had Hashimoto's for over 10 years, and my anti-body levels are still elevated everytime. My synthroid level is at 185, and has been for about 2 years, but we'll probably be moving on up higher before long. I also have to take cytomel at 20mg a day, to keep the t4 (??) level up (I can't remember what's for what hardly anymore... sigh). Anyway, since I've been on plaquenil, I was hoping for a reduction in the levels of *all* antibodies in me. No such luck thus far, but I go next Monday the 7th to the rheumy for more follow-up, and we'll see then.
If your gp doesn't want to do it, ask the rheumy to do it. Do the *full* battery though, including fasting lipids, so they get an accurate picture. Just doing TSH and T3 doesn't give the whole picture. You need a couple others (the names escape me at the moment... sigh). Be sure and have the rheumy share the tests with your gp, and have the gp share their tests with the rheumy. If you don't tell them everytime it seems, they tend to forget...
"There but for the grace of God, go I."
"... His mercy endureth for ever."