You are right and have learned a lot about your child's problem including not using ice on anyone with cold-induced urticaria (hives from cold).
Cold urticaria is a relatively uncommon physical urticaria. An urticarial rash develops after being exposed to cold, cold water, and cold objects. Swimming in cold water may cause severe wealing and fainting, and should be avoided.
If she is in a fair amount of pain you should call and talk with your Dr. to see if she could try a children's nonsteroidal anti-inflammatory drugs (including ibuprofen). Some DRs. do not let children with cold urticaria use NSAIDs.
I hope she is soon feeling better.