Mike's link is a great one. I'm sure you found much there to relate to. I'm very interested in your story.
You wrote that your daughter has PCOS. That is a clear-cut need for elemental iodine, found in either Iodoral or Lugol's. Simply put, PCOS is for the ovary what FBD is for breasts -- cysts in either the ovary or breast that respond to iodine. As a target, she needs to aim for at least 50 mg of either Iodoral or Lugol's, but with the Hashi's, she might want to start low and move forward when she senses it's time. Since she's gluten intolerant, she might really benefit from no grains whatsoever to lower inflammation in her body. The companion nutrients will provide her with antioxidants. A full-spectrum Vit E would also be useful, maybe 400 mg. Depending on her free T3 and free T4, she might need natural dessicated thyroid hormone as well. fT3 needs to be at the upper end of the range, and fT4 at least in the mid range. Another really good website is Stop the Thyroid Madness http://www.stopthethyroidmadness.com/
I think what the medicos have put you through means you need to be very careful. At least, I don't have clarity about what you need to do. When you said your T3/T4 were normal/ok, did you mean fT3 and fT4? These are the only numbers I have a passing familiarity with. My story - my thyroid did enlarge on iodine, twice. Once many, many years ago on about 30 mg, but my doctor felt the enlargement, and I knew I'd been experimenting with the kelp/iodine I could find in the health food store, and I was scared of getting too much attention from her. So I stopped the iodine and my thyroid got smaller -- I will not say it became normal. The second time was after I found this forum and began Iodoral and SSKI. I think the iodine put pressure on my thyroid to perform, and it couldn't. I tried and tried to make iodine fix my thyroid, but it just didn't happen, and I had to start Armour thyroid. Now my goiter is almost gone. Seemed like once the thyroid was performing more normally, it could take the iodine in to decrease the goiter.
Not everybody has this dilemma, but I wanted to let you know that your path may zigzag quite a bit. I believe with your the knowledge you are gaining you will be able to sort through your options.