Can I suggest you write a polite letter to the vet and thank him for his care and concern for the cat. You can then go on to explain the purpose of your organisation, the services you provide, the limitations on your resources and suggest that if he wishes to provide emergency treatment for an animal in the future you will welcome him contacting you for assistance - before the fact - and may, in some circumstances be in a position to assist.
As others have said - if he wanted payment for the services, the person who adopted the cat should have provided that payment, if indeed anyone other than the vet paid for it - he acted on his own recognisance and is therefore liable for the costs. The cat is probably rehomed to a close family friend and he feels uncomfortable raising the issue with them ...