Emily,
if the choices are breaking the contract or knowing that the dog will be put to sleep if he goes back to the rescue place, breaking the contract and finding a new home for him seems to me the most decent thing to do. Dogs can adjust very quickly to changing circumstances when they are given love and attention and taking him to a new home shouldn't be the concern in this case.
I wonder how much investigation, home visits, etc., this rescue does before they allow a dog to be adopted. I do volunteer work for Yankee Golden Retriever Rescue and know that it may be easier to adopt a child than to adopt a dog from us, considering all the steps that prospective adopters have to go through. Also, I never heard of one of our Goldens face the possibility of being put to sleep if he/she goes back to the kennel. If the dog's history is not know, because of being a stray, then that dog is not given to families with children.
Regardless of how well intentioned these people are, I do hope, for the sake of any other dog, that they will give up the idea of adopting a new one in the future.