If you stop payment, the likely outcome is they will put the bill into collections. You might contact the first vet, explain what's going on, and ask them to waive the bill or refund your payment.
I would start with the vet and explain what happened and see if they will negotiate the bill down or waive it. If you just stop payment on it, you could end up in collections.
I would wait to see what the outcome of the surgery is before contacting the other veterinarian. Has this other veterinarian jeopardized your dog's chance of recovery? Depending upon the outcome, you may have a larger claim against the first veterinarian.
If your dog recovers just fine and the only issue is the first veterinarian's bill, then depending upon the size of the bill, you may want to have an attorney write a letter to the veterinarian explaining why you believe that you do not owe it. If the cost of hiring an attorney is higher than the vet bill, I would write the veterinarian and explain why you feel that you do not owe the bill. If the veterinarian does not waive his bill, you may want to take him to small claims.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.