I'm so sorry your question has gone unanswered for this long. I hope you haven't given up on us.
I am hypothyroid, and my doctor is a homeopathic MD. Even he treats low thyroid with prescription thyroid hormone. I truly doubt your hyperT can be treated by homeopathics alone.
I don't blame you for not wanting the "drastic" treatment. (I assume you mean RAI ablation?) I wouldn't accept that, either. Not without giving anti-thyroid drugs a darned good chance at it. And if they didn't work, I would opt for sugical thyroid removal... never RAI. Do your homework before you agree to that.