Paizly went to the vet tonight for her shots and I did ask what they would do in this situation. The girl I was talking to said it has happened once while she was there and she freaked out and started yelling toward the back for help. Partly because she was in the reception area and the dogs were in the dog area. There is a counter between. But she is also a cat person so she had no clue.
But I told her this story and she said she was definitely going to tell the story and see about having a plan of action. I'll probably ask (if I remember) each time I go in there.