"Could it have something to do with faulty permeability of cell membranes or transport of Iodine or something else into/out of cells?"
-definitely. I cant see anything working well if the digestion of fat doesnt work. Permeability of cell membranes,yes.
- think sulfur; http://www.iamm.com/gi-index.htm
"lipase/phosphatidylcholine"
I had great effect of taking enzymes in the beginning of Iodine supplementation. My digestion really was no good.
When it start working, as seem to be in your case; utilization of fatty acids will increase, hence everything work better.
To have an impact on the thyroid; saturated fats like coconut oil are a hit. But only if one can digest it properly, of course.