As already suggested by Jsusnme, 26 months earlier, try Iodoral tablets. According to Optimox, the maker of these tablets, colloidal silica is utilized in the tablet specifically to avoid gastric upset:
"Because administration of I[odine] in liquid solution is not very accurate, [it] may stain clothing, has an unpleasant taste and causes gastric irritation, we decided to use a precisely quantified tablet form containing 5 mg Iodine and 7.5 mg Iodide as the potassium salt. TO PREVENT GASTRIC IRRITATION, the iodine/iodide preparation was absorbed unto a colloidal silica excipient; and to eliminate the unpleasant taste of iodine, the tablets were coated with a thin film of pharmaceutical glaze."