Yes, there is a step in thyroid hormone synthesis that requires molecular iodine. Your body is supposed to make it from iodide, but there can be hiccups, as you've found out. To get from iodide to iodine requires niacin for organification, ferritin of 90-110 that is somehow involved with TPO (unclear to me), and salt and vitamin C for the NIS (sodium-iodide symporters). I myself had a boost in thyroid activity when I increased my elemental iodine with Iodoral (or, for you, Lugol's). Take the Lugol's straight, no more in orange juice. And take it every day. As you've seen, even taking it a few times a week your thyroid still aches, so get the benefit of daily Lugol's. Get your ferritin up, which could take months of taking iron. Don't underestimate how much that could help. And I just ran across a note I'd made, that Dr. Flechas said that autoimmunity is a defect of methylation. When you increase methyl groups, such as by taking trimethylglycine (TMG), your antibodies drop.
I also belive that even if you hadn't put the Lugol's in oj, you still might have ended up with high antibodies since you have other holes in your thyroid hormone synthesis that would have eventually shown themselves. Fix these holes, reduce your inflammation (no to gluten, yes to trying TMG) and your thyroid will heal.
Oh, and one thing iolite mentions quite a bit -- getting on enough dessicated thyroid such as Armour, Erfa, will also cool the inflammation. All the iodine and supplements in the world did not tame my goiter, till I added dessicated thyroid, then everything fell into place. In the meantime, I'd raised my ferritin, taken B2 and B3, etc., etc.