I was diagnosed with Hashi 2 years ago at the age of 61. It was first picked up by my allergist who noted the facial and neck rashes I had been complaining about. He ordered the antibody test and it was very high. I had multiple tests (blood, scans, and biopsy). I continue to be monitored, do not yet require thyroid medication, and still experience bouts of the rash, for which I use Mometasone Furoate Cream. My allergist told me that he has a number of patients who were diagnosed with Hashi as a result of having an associated rash (not cause and effect). I don't know if this info will help you and you daughter, but I wish you well.
Hashimoto's thyroiditis is pretty common and often leads to autoimmune hypothyroidism, sometimes the thyroid can become overactive and we call this Hashitoxicosis.
Rash usually is not related with ongoing well treated hypothyroidism.
There may be a rare reaction with her medication but if she has been stable I doubt this is related. I would first check with her pediatrician.
I would address it first with your ped, then call your ped endo to see if she needs a lab test to assess the level of hormone in her system, obviously overreplacement can lead to elevated thyroid levels which can have related symptoms but not usually just a rash