I have supplemented with lysine earlier; couple of years ago; didnt have a benefit I know of.
But I will try if I get cold again; had a period of bromide detox when coldness came for a revisit. OK now, after upping the Iodine once again. (probably helped that the temp outside went from -15C to lukewarm
Many forum members have reported coldness when taking iodine; but this is a trancient phase; dosage matters.
I remember you are in the sensitives league as I am. We react easily to whatever supplement; and IMO we are vulnerable when doing to many things at once.
I have found the simplest protocol to work best with me; iodine, salt, boron,selenium, magnesium, vit c and b. I have to be careful with b-vit though.
For me coconut oil is the best for warmth and steady energy, in addition to iodine.
It is a puzzle to find out of ourselves; I would go for what works; like in your case lysine, and Iodine that seemed to work for you earlier.