I say ABSOLUTELY dispute. Take it up with your bank if they do not concede. Most banks provide the information on how to go about this on the bank's website. I had a similar experience (although it was not a cat or a vet) considered doing the same, but only due to procrastination I did not submit the dispute claim. Something I greatly regret.
You appear to have the energy for it. I'd decide on a strategy, inform them, and have at it no holds barred.