If you do not seem to have a problem with seaweeds, I would find good clean sources and use them daily. Get creative in their use with your food preparation. I really like dulse in my foods.
Or possibly the magnascent Iodine mentioned below. I know nothing about the Magnascent iodine. High on Water uses the Iodine at iodine source. Perhaps she may have some words of advice on it for you.