that is very frustrating. I am sorry you had to go through all the red tape, and still get denied by A vet error. There is a new vaccination shedule and if I am correct they are every three years now. I am not sure about the Rabies vaccine though.
Rescue groups are very, very selective and IMO too picky. My parents were turned down over something about the vet too. From there last dog not having A blood test. All my parents dogs have lived to 16-17 yrs old . And are wealthy, retired couple. they did adopt a cute Lapso Alpso mix , eventually.
Don't give up! Keep trying , and do go to your vet office and nicely get copies of your dog records, even discuss it over with the vet, and make sure everything is in order. That is if you are going to try getting a dog from a different rescue group.