My regular endo takes pediatric thyroid patients. Don't limit yourself to a pediatric endo - most of them are dealing with pediatric diabetes. Call and find a good thyroid specialist endo then talk to the nurse (a secret to getting around the receptionist) about your daughter's condition - that may get you in faster.
Starting her on Synthroid isn't a bad idea now but she needs long-term followup with an endo. While her primary care doctor is starting this, did they do an ultrasound of your daughter's thyroid or just determine the multinodular goiter through palpitation? If she hasn't had an ultrasound, she needs one.
She needs to be on meds to bring her TSH level down. Trust your doctor. Wthout the meds, she will continue to go more hypo and this can be devastating She is not that far off of the normal range, so a low dose would be suggested to start with, even if it is 25mcg, but get her started on it immediately.