You have not replied to my information request, but I am going to assume that from your original fact pattern, the promise was verbal.
If the promise was verbal, then it would be very hard to prove. They can simply deny that they ever made this promise. In addition, it was not binding because you did not pay anything for the agreement for them to hold the other kitten, called "consideration" in contract
doctrine and generally required for a binding contract.
As such, I am afraid that there is little action that may be taken against the clinic. I am sorry.
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