I agree with your very legitimate worries about radioactive iodine and I am not at all sure of the logic underlying the clinical advice you have received from specialist clinicians, your endocrinologist and oncologist. 'You already have CML, so you can't get it again'? to this, I would say that CML is not something you 'get' then 'get rid of' then 'get again'. The respective clinical 'status quo' may well support this kind of thinking, but as I said, I question the logic.
There is little to no research, at least that I can find, that would make you confident that an increased exposure to any kind of radiation - including radioactive iodine - would at the very least not pose a risk to the control of CML via TKI therapy.
Since my own CML diagnosis (1999) I have consistently refused to have and kind of radiotherapy or radioactive anything -including X-rays, mammograms etc. (this is for another discussion). Some would say my attitude is based on an irrational fear and non-scientific thinking. I choose to go with my gut feeling and my own research.
It is accepted that this is a link between exposure to ionising radiation and the development of leukaemia, including CML. A direct link to CML might be contested by some, but for the Japanese citizens and others who were exposed to overwhelming levels of ionising radiation from the atomic bombs dropped on Hiroshima and Nagasaki at the end of ww2, the fact that they caused the development of leukaemia is not disputed.
Regarding thyroid problems, there is some evidence that tTKI therapies affect the thyroid gland (with possible consequent sub-optimal thyroid hormone levels and its symptoms). If you already suffered from hypothyroidism and had developed nodules on your thyroid gland, then it may be (according to my own research on behalf of my mother who suffers from hypothyroidism and had a nodule 'treated' via a partial thyroidectomy at 28 years of age) that you could try to support your thyroid with iodine. You can by this online.
My mother has recently resolved her issues with yet another nodule on her thyroid (non-malignant) by including 6 drops of iodine tincture in water twice daily to her routine. She has seen the swelling in her throat and symptoms related to that, disappear over the last few months since adopting this routine.
I suggest you do some more research into the causes of thyroid nodules before you succumb to surgery. You may want to join the UK online group for hypothyroidism and Hashimotos Thrydoiditis at www.Healthunlocked.com