Read the comment at the bottom by Puppyluvnj in the following thread:
http://www.medhelp.org/posts/Thyroid-Disorders/Diagnosed-with-Hashimotos-AND-Graves-disease/show/944756
She went into remission for 20 years...that's a long time. Most often meds are tried first and a "permanent" solution is only recommended when/if meds fail. Before considering surgery or RAI, I, personally, would want to see what my reaction was to ATMs (anti-thyroid meds).
It's true that thyroid replacement hormones are not drugs, but substances our bodies make naturally, so there are very few side effects to the meds themselves. Be aware, however, that it's not always quite as simple as your doctor is making it sound...read some other threads on this forum and decide for yourself whether there's "not a chance of reaction". A low incidence of side effects, yes...but there are challenges.
I'd get a second opinion before doing anything permanent.