Ideally, your ex-girlfriend would come forward to pay the balance, but apparently she will not do so?

You should be able to bring an action against your ex-girlfriend within the context of the litigation, arguing that she should indemnify you for any judgment against you.

You should also be able to defend against the action by arguing that there is no contract between you and the vet. That is, although you permitted your girlfriend to pay part of the vet bill with your credit card, she was not authorized to use your name when registering for the services, and you should not be held responsible just because the vet required her to register under your name instead of her own to facilitate their billing your credit card.

I hate being this theoretical - obviously, you would benefit from consulting in person with a New Jersey lawyer, and getting specific advice and recommendations.