I hope you get some news soon! When you mentioned that about iodine, it got me thinking. I have a form of Celiac Disease called Dermatitis Herpetiformis - it's a dermatological manifestation of CD, and iodine will trigger a rash in some people with DH.
Your symptoms certainly sound like MG, but if you have one auto-immune disease, you are more likely to have another one. You may want to ask your doctor to test you for Celiac Disease while he/she is running other tests. Be aware, though, that about 10% of people with DH will NOT test positive on bloodwork for CD (I'm one of them) - they still have it, but the antibodies are expressed on their skin and not in their blood.
Many people on this forum have benefitted from a gluten-free diet - it won't cure MG, but it does seem to lessen some of the symptoms. A lifelong GF diet is currently the only known "cure" for Celiac Disease/Dermatitis Herpetiformis. People with DH are advised to limit their iodine intake - that can be done fairly easily by using uniodized salt at home when cooking.
Here's a link to more information:
http://neurotalk.psychcentral.com/post8767-2.html
Hope this helps!