I don't have a dog with hypothyroidism, but I myself suffer from autoimmune hypothyroidism. It is difficult if not impossible to get any answers from a doctor to my questions of what causes a flare up (autoimmune version comes and goes, if you have an test for autoimmune antibodies they have to test you various times, as the first test can come back negative which is what happened to me and then two months later it came back positive), all the doctors do is stick you on thyroxine and forget about you except to have a yearly blood test.
I had a very bad time with the thyroxine, apparently the dose was too high for me (although it is the bog standard starter dose of 50 mg for adults and the side effects were muscle pain, tiredness, headache, earache and rapid heartbeat), I eventually went to a sympathetic doctor & nutritionst (private health) who put me on a minimum dose of 25 mg and changed my diet to exclude all food with gluten and to include a regular amount of fish into my diet. Thanks to my osteopath's suggestiosn I also supplement with magnesium and omega 3 fish oil. I feel better than ever now, and my Spanish NHS doctor is at last happy to let me carry on at a minimum dose for the time being.
I have read quite a few books about autoimmune hypothyroidism and the americans at least seem to think that it's gluten causes the autoimmune response to flare up. As I said national health doctors really don't want to look into the cause as the treatment is so cheap and easy, so I don't think there would be many veterinary studies about it either. so for what its worth it might be an idea to check out there is no gluten in your dog's food or try a salmon and rice variety. Also thyroxine treatments need regular blood checks to see the dose is right, in humans it is at least once a year.