Is the hypothyroidism autoimmune based (ie: due to antibodies attacking the thyroid glands) ? A blood test can indicate if there are active antibodies.
If so, then the condition, despite being supplemented with medication for low levels of thyroid hormones are possibly only addressing one issue. If the medication is only supplementing T4, then T3 will still be compromised. A blood test can be telling regarding optimal levels and indicate if T3 needs supplementing.
Secondly, if the disease is autoimmune based, then the dog’s body will be under continual inflammation. My suggestion, is to scrutinise what other external impacts are involved to remove the inflammation. They could be other medications (pain medication – such as opioids – are great for pain relief, but trigger and maintain a key inflammation pathway in mammals; so replace pain meds with non-opioids). Others include diet. Some foods are highly inflammatory. Obvious ones are those with high levels of glycoalkaloids (such as nightshades), sugars and some grains (particularly those with gluten, lupin and corn).
Other things to investigate which often go hand-in-hand with hypothyroidism are ferritin (iron) levels, vit D and B12 levels. You can check these via a blood test as well.
I hope you can get some positive solutions in place.