I found your post really interesting. I had a total thyroidectomy 9 months ago. I was 23 at the time and had suffered with Graves for 3 years. My story was slightly different to yours in that I was never well controlled and never went into remission, and had to have RDI or surgery before finishing Carbimazole.
I completely understand your comments about not wanting either procedure, and I think it's worth looking in to natural treatments. However, I have to say that I personally tried every single thing I could find and nothing worked, in fact, a lot of them made me very ill, so treat them with caution. Especially if you decide to stop taking Carbimazole.
I personally chose surgery over RDI as I was 23 at the time and had heard RDI can affect fertility (in females), with that already in question after a long, uncontrolled period of Graves, risking it wasn't an option for me. Whilst I don't want children now, I do someday. I was also warned that RDI is not as effective in cases of extreme hypothyroidism, as mine was.
I won't lie, surgery wasn't a walk in the park and it hurt, but it did stop all my symptoms.
I know you don't want either option, but you should consider them regardless, the fact is, they might not be wanted but they may be necessary for your health and well being.