First of all if your doc isnt interested in T3 or T4 levels you need to find a new doctor.
Secondly, you need to decide for yourself if you are going to self treat or see a doc, there is little room in the middle, especially if you arent going to be forthcoming with your doc and tell them what you are taking.
Your TSH should drop after you reach Iodine sufficiency. How long is that? Largely depends on how deficient you are and how much your taking.