I can tell you what I do.
I self medicate with NDT instead of Levo. Levo did nothing for me but with NDT I am well. And I keep my bg normal through LCHF.
When some of the symptoms of hypothyroidism started to creep back I couldn't understand how this could happen. It was diabetes though, terrible tiredness, lousy memory, pins and needles and cramps, difficulties to get going. Day of diagnosis I went on a strict low carb diet, kept the same amount of protein but upped my fat intake. Week after diagnosis I got a bg monitor and strips and have been testing to see what I can eat and still have normal, and I mean normal normal bg. It works very well. No bread, no pasta, no potatoes, no rice, a lot cheese, butter, olive oil and vegetables. I have lost some weight in these almost three months, not a lot but I don't need to loose much and I have the same level of energy I at twenty-five so long as I stay off carbs and keep my bg normal, preferably below 7.
So my advice is to go low carb and up the fat intake to control bg and possibly discuss other options in thyroid meds with your Dr.