First, you may have a case for negligence. The definition of negligence is causing injury (monetary or otherwise) due to a breach of duty. Did the vet have a duty to handle your dog with reasonable care? Did the vet breach that duty? As a result of such a breach, did you suffer an injury?
Second, the fact that you took the $350 check does not necessarily mean that you are "even" with the vet. If a court determines that your dog is worth more than $350, obviously you would be entitled to the difference if you won a negligence case.
The one good piece of advice you got is to consider the expense associated with litigating the matter. You could wind up spending quite a bit more than you might win.
The ONLY comment I'm going to make regarding your post is that if you knew what you were talking aobut you'd be an attorney. The court may in fact determine that the dog is worth more than $350. That DOES NOT however, mean that the poster will have a chance of recovery. And to suggest otherwise is to act irresponsibly on this forum.
The poster has gotten the same advice I would give her had she sought me out and sat in my office.
As for you Turbo, if you feel my advice is not relevant you are (and should) consult with more than one attorney and ask not only the points I gave you, but also the cost of such litigation, the proof involved and the proximate cause of the loss including your associated liability.
IF, after all of that, you decide to continue to suit, that is your right. But even the most competent attorney cannot tell you that you WILL win or you DO have a strong case.
As I said, in my opinion, the vet HAS been harmed. She doesn't have your business any longer.