Do you have any root canals or teeth that are under a cap or bridge?
I've this theory that Iodine will oust bad teeth or make their presence known, however, it's not always easy to tell it's a bit of detective work unless the obvious where you go OUCH!
I've had similar issues to what you are describing and it was root canals gone bad, my body rejects them, anyhow the minute I would take my Iodine up then I would swell and even my thyroid meds wouldn't touch this swelling..I was sure it was NOT any of that because I swelled nowhere else on my body just my face/eyes. It is something to think about.
And yes, the companion nutrients are really important.
The only other thing I can think of causing swelling would be kidneys but you would swell over your whole body. Liver would cause legs to swell and under eyes if really bad but I think you would know it at that point if your liver was in trouble so that rules that out.
Swollen face is not fun. I hope you figure this out. It is quite possible you are allergic, it's hard to say what exactly allergies produce in different people but swelling is definitely one of them, is your tongue swelling? That would concern me if your tongue is swelling. Have you tried seaweed or kelp in place of the iodine?