Having gone through something similar, I can tell you that fluid retention is usually due to lymph congestion/stagnation. Corrections include using things like chi machines, tens units, ultraviolet light wands to move lymph or sauna to make you sweat.
Thyroid patients lose the ability to sweat effectively. My experience was that I would just get really overheated and swell up like a boil-in-bag... Hence the added weight gain. So, if your body is starting to warm up (from the Iodine) but has no way to eliminate the heat/water, you're going to retain it.
Learning to sweat again is hard work. You have to raise your body temperature -- by several degrees. And you have to do it safely. I don't know what your personal health stats are (BP, BMI, etc.) so PLEASE just use this information as a place to start...
You can look on line for plans to build your own Far Infra-Red Sauna.
You can purchase parts from Home Depot - Home depot sells clip-on task lights for about $15.00 and an Infra Red light bulb (used to heat bathrooms and amphibian environments) is about $8.00). Providing you can carefully array several by clipping them under a dining room table (the old-fashioned kind that has leaves and wooden lips than hand down far enough for clipping, take a bunch of blankets and throw over the top and sides of the table, and then put some towels, blankets on the floor. Crawl into your sauna and stay for as long as you can stand it – not to exceed 30 minutes. You need a thermometer so that you can keep track of how warm it gets in there as well as one to check your own body temperature.
You can buy a kit.
You can buy a ready assembled one – like Solo-Spa. It’s expensive. But – I lost my water weight (and a whole pants size) very quickly. And, it comes with a body-length FIR (Far-Infrared-Sauna)
heating pad that feels heavenly for the muscle aches and pains that often accompany thyroid conditions (e.g. fibromyalgia).
If you are not drinking distilled water enriched with balanced minerals – that is also very important. Each mineral has a specific atomic weight. If they are out of balance- they can’t push the others out – so your sodium gets out of whack. Important to track your minerals. Again - I am not a physican - need to work with your doctor to deal with the mechanics.