Find another Vet! That is horrible treatment, especially for a puppy. The only reason your pup may get aggressive is if it keeps getting treated poorly at the vet.
We also have 6 cats and whenever we bring a cat to the vet, Molly wants to come - she gets very worried. So the vet tolerates Molly and the cat in the exam room together, she always has a nice greeting for Molly and said that Molly's behavior has improved so much just allowing her to come in as "support" for the cats. Hope you find a vet practice that has more positive practices.