Wow, that is frustrating! Hopefully, others here will chime in. I’ve heard of hives with hyperthyroidism or with antithyroid meds, but your situation is obviously different.
One thought is that I’ve heard a few anecdotal stories of patients reacting to fillers/binders or dyes in their thyroid hormone medication. It would be worth reaching out to your doctor. For people who have this issue, there is usually one dose that has no dyes, and you can adjust the number of pills accordingly. There is also a levothyroxine product called Tirosint that is a gel capsule (and also available as a liquid).