I don't want to hijack this thread, but just share my experience in case it's of use to anyone.
I took Lugol's about daily for maybe six months for my classic sluggish thyroid, etc. Really I was very happy with the results. My temp climbed from 96.-something to mid-98s, and bad breast pain and itching resolved. My blocks-of-ice feet at night disappeared.
But not too long ago, I noticed my period had dwindled to nothing, which never happens. About a month later I did a 7 day fast during which hyperthyroid symptoms ramped up fast and dramatically (I didn't recognize them, knowing nothing about hyper). During this period my thyroid swelled hugely (diffuse toxic goiter, which I still have). The symptoms got worse and worse so I hit the MD with high blood pressure, heart racing out of control.
The MD thinks I have Graves, but of course what I have is excess iodine. Unfortunately, I also have begun manifesting thyroid eye disease. (Note to doctors: this is NOT limited to Graves!).
This is a difficult change in my life. I am taking beta blockers which in England are basically banned (connection to diabetes) and heavy doses of tapazole. My thighs are weak, my hands tremble constantly, imsomnia is common (before being treated I would stay awake for 48+ plus). The hyper is causing my heart to work too hard all night and day.
My understanding is that my hair will start to fall out on the tapazole, and the eye disease already looks strange and is verging on disfiguring. There is no set treatment for the eye disease -- radiation treatment and a somewhat severe form of surgery are some of the options for resolving it, after it has a chance to run its course (several years). I feel better after taking the tapazole but typically don't do well on long courses of drugs ... sissy liver.
It's possible that this will all resolve sooner than typical for Graves patients, since it's due to the iodine. But it's looking very possible that I might go through the whole Graves-like type experience.
Note that this type of reaction to Iodine is NOT universal, obviously. There is a subset of people who are not able to process too much Iodine correctly, like me. The problem is, if you're self-treating hypo symptoms, then it might be more likely that your thyroid falls into that not-quite-normal camp. So I would be cautious, and familiarize yourself well with the symptoms of hyper before you go too far. (Of course, your malfunctioning thyroid might also decide to go hypo in response to iodine).
Hopefully, fingers crossed, I will post back in a few months of not taking iodine, reporting a miraculous recovery. Be well, everyone.