I doubt magnesium oil would be enough, although it has its uses. We generally recommend oral magnesium citrate or magnesium malate. You need a minimum of 400 mg of the magnesium component, up to around 1200 mg according to your bowel tolerance before you get diarrhea.
I think chlorine is nasty to the body. I gasp for air around chlorinated swimming pools, and my husband got bronchitis the last time he took a swim in one. I think you could be right, that this is the source of your trouble with iodine.
If you're so inclined, you can test for RBC magnesium. A useful target suggested by cardiologist Dr. William Davis is >=6.0 mg/dL. Red blood cell testing serves as a predictor for tissue levels, as opposed to the level in just the serum. Test after you've been using a generous amount for at least several months. As Ginagirl says, it takes a long time.