We have a hyperT cat, diagnosed early last fall. His bloodwork suggested he was possibly diabetic or becoming diabetic, so we started feeding him as if he was, and now he eats Fancy Feast classic pate, exclusively. His questionable blood sugar numbers went down into normal range, so we are continuing to feed him as if he is diabetic, hoping that by removing the stress on his body of processing too many carbs will help keep him under control (for diabetes).
The only diet that theoretically might help with hyperthyroid is one devoid of all iodine. But it rarely works, and it generally doesn't last very long. Theory is that without iodine, the thyroid can't produce its hormone. But, a hyperthyroid cat has a tumor (almost always benign, not cancerous) growing that produces the extra hormone, and as the tumor gets bigger, it makes more hormone and the cat gets more hyperthyroid. Lack of iodine MAY slow this process for a bit, but it won't stop it, and it won't cure it. Special diet thyroid food is designed without iodine. But not for anything else. The high carb content puts the cat at greater risk for becoming diabetic than it already is because of the thyroid problem. It's certainly not healthy or balanced for what a cat needs. And it probably won't help with the hyperT, certainly not long-term.
A hyperthyroid cat needs to be on medicine to reduce the hormone levels. There are pills, and there are various forms of a cream that you just smear inside the cat's ear. We use one of those, comes in a pen you twist to get the dose squeezed out onto a soft applicator tip, then we just rub it down inside the cat's ear, very simple. He doesn't even care anymore. Got used to it very quickly.
Here's a link to a very good facebook page for hyperthyroid cat owners:
https://www.facebook.com/groups/42301610484/ Great place to ask questions and get advice.