The doctor could do a thyroid ultrasound and check for a nodule that way. No iodine is involved.
Your FT3 and FT4 are both barely within reference range. This clearly indicates hypothyroidism. However, your TSH is also quite low. TSH is counterintuitive...when you are hypo, it's usually high, and when you are hyper, it's usually low. The low TSH with low FT3 and FT4 might indicate some pituitary involvement. TSH is the least important of the thyroid tests since it measures a pituitary hormone, and not a thyroid hormone. I would ask for antibody tests (TPOab and TGab) to confirm or rule out Hashimoto's thyroiditis, an autoimmune disease. A thyroid ultrasound might also be in order. Given your allergy to iodine dyes, I think you might postpone (at least) the scan. I'd try thyroid meds...I think they'll make you feel a lot better.
I would highly recommend to ask this to Dr.Lupo here on ask the doctor forum ....did you check both antibodies ?