First of all, the human body requires Iodine to be healthy, and 3 mg a week is such a trivial amount that many would say that it is not enough. I take 25mg a day for my prostate & my wife takes 12mg a day (iodine with selenium) The selenium is required for thyroid health. I think your docs are over-reacting for no good reason. What was her blood level before taking iodine? There must be another reason for her blood level & I doubt that eliminating Iodine will fix it.
Iodine is especially required for women to prevent severe problems with breasts, ovaries & uterus/periods. Fibrocystic breasts & related breast cancers are cured with iodine. In mt home town, we have cured a woman with fibrocystic breast disease & another woman with terminal uterus cancer with 50mg of Iodine daily with the required co-supplements. See my research link here: https://sites.google.com/site/iodinereferenes/home
Hope this helps & we need to know which blood level they are worried about.