I don't have Graves'. I'm at the other end of the spectrum, Hashi's hypo. But I've done a lot of reading about Graves' and know what I would do if I had it. I would NOT do RAI. In case you haven't been told, you need to know that your risk of eye damage from rampant antibodies increases if you take RAI. You risk becoming even more hyper in the weeks/months following the treatment. There is an increased risk of various cancers in the years/decades afterward.
I would find an endo who believes that anti-thyroid drugs can put the disease into remission. Most will push you into RAI, because that's easiest for them to manage. They believe that being hypo is preferable to hyper. Those of us who are hypo can tell you... Endos are the pits at treating it, too. It's not the walk in the park to treat that they would have you believe.
Tapazole can be tricky to dose, but if you have an endo willing to go the extra mile to make it work, you stand a very good chance of going into long periods of remission without being made permanently hypo.
Speaking for myself, if anti-thyroid drugs didn't work, I would opt for surgery at the hands of the most experienced thyroid surgeon I could find. Yes, that would cause permanent hypoT, but at least there would be no risks or side effects of the radiation.
Good luck to you in making the right decision.