First part here:
A carnivorous diet poses no threat to anyones thyroid. In fact if you are hypothyroid it's likely that you already eat too many carbohydrates, seeming that most cases are autoimmune or stress related in origin. Carbohydrate consumption in the form of easily digestible sugars and gluten containing grains have a negative impact on the thyroid and adrenal glands. My recommendation would be to minimize or eliminate them completely
The "Seven Deadly Sins"
Grains (wheat/rice/oats etc) . . . . . Dairy (milk/yogurt/butter/cheese etc) . . . . . Nightshades (peppers/tomato/eggplant etc)
Tubers (potato/arrowroot etc) . . . Modernly palatable (cashews/olives etc) . . . Refined foods (salt/sugars etc )
Legumes (soy/beans/peas etc)