Well the Mainstream consensus is that Iodine makes a Hyperthyroid more Hyper in a way. But I am hearing that it's good for both HyPO AND HyPER. Check on that over at the Iodine VWT Forum and ask about it there. They have more info on it.
Several years back I was diagnosed with Hashimoto's Thyroiditis, which is almost like an autoimmune Hyperthyroid and the Doctor at the time told me it would burn out into a Hypothyroid condition and to start taking Thyroid Meds immediately. I didn't like the way The Meds made me feels so I took about 2 and quit.
A few years later I started on Lugol'sIodine and I could not do very much at first. Then I slowly upped my intake. Then I dropped my intake back down and I take Iodine but less, and I take breaks. Btw the Mainstream consensus for Hashimoto's is to stay away from Iodine supposedly because "Iodine causes Hashimoto's"
Anyway as of a month ago my blood test shows NO Hashimotos and my Thyroid is perfectly normal.
So I'm thinking that if Iodine can help Hashimotos it can help a Hyperthyroid as well.
I would try Iodine with your Dog but start out with very little and skip days sometimes. She may not even be Hyperthyroid. Even if she did have a tumor it would help with that as well. Maybe back off a little on the Enzymes and feed her more meats and see if that helps. Some Dog breeds need more protein than others.
Doctors do have their place, but I'm not sure how reliable the tests are especially if the numbers keep going up and down. You may be just spending a lot of money unnecessarily.
But after all the suggestions keep an eye on her weight and if that continues to be a problem then I would do further tests.