I would have told him right then and there, "Do not call my child a brat again", and gotten his name, and reported him not only to the hospital, but to the medical board or whatever.
It may not be illegal or malpractice, but it is rude and could be considered abusive to call patients names.
My middle child has Autism. He got sick a couple of years ago and could not keep anything down for a couple of days, so I took him to the ER. I did explain, when we came in, that he has Autism and may need extra help for IVs or whatever.
He screamed when they wanted to put an IV in. One nurse attempted to get snarky and I politely said, "Ma'am, my son has Autism and he is scared to death. Let me help you hold him and you will get the IV in." She backed down, was nice after that, and the rest of the experience was pleasant. He was moved to a room with a TV and we watched cartoons while he got his IV fluids.
If she had persisted in being rude to my scared, and disabled child, I would have reported her to whomever I could.
I have worked in hospitals, drawing blood for the lab. It is a hard job and sometimes you have to deal with really crappy parents. But, I would have NEVER been nasty to a child because of it. And for the record, I treated even the crappy parents with respect.
If you cannot handle the job, QUIT!