I have had Graves for 2 years, took 10 mg Tapazole, now reduced. I added acetyl-l-carnitine, 3,000 mg and had amazing positive results. See Google "L-Carnitine in the Treatment of Hyperthyroidism".
That is a popular misconception that the thyroid will eventually ''burn itself out''. It's only true for less than half of graves people, and these are the ones that most likely test + for the hashi antibody(TG) as well. Graves stimulates the thyroid not destroy it, much like hashi's does. Sounds like your endo is pushy. Many many many ppl have been taking ATD for 20-30 yrs with no negative effects. I'd rather take ATD's for the rest of my life, (and or go into remission here and there only to start ATD's again) and keep a vital organ, than just get RAI(please read up on the side effects) to please a doctor. You'd be amazed at how many chose RAI b/c they felt pressure from their endo. The deciding factor needs to be YOURS and YOURS alone. On the flp side, if the meds aren't working, I'm all for looking into a TT!
Your hair loss may be due to having low iron, which most with autoimmune thyroid disorder have. Many times symptoms have a secondary cause,
When was the last time they did a CBC, or checked your vitamin/mineral levels?
Its a decision you need to make carefully.
I'm concerned here alot with you.
On a TSH of 1.34 - that sounds pretty good, even though basing everything on a TSH lab - AND pushing for RAI - soley on that - is very wrong.
Symptoms you mention point to further testing prior to permanent ablation and you need to have the Free T3 and Free T4 labs done ASAP to see how your body is reacting to the anti-thyroid meds currently.
His blunt direction is somewhat appauling and is telling me he doesn't want to do his job more in depth and possibly lazy - to push for RAI and not try and help you keep your natural function or do any healing first to see if you can come out of this better.