E.B.
This isn't the same situation, but perhaps it might give you an idea. My son was thinking of adopting a particular dog from a shelter. The shelter allowed him to take the dog to a vet, where we had a thorough exam. In this case we wanted a vet to give us her opinion about how the animal had been treated, and if the dog had any potential problems. The vet confirmed that from the way the dog interacted with her, it had been raised lovingly. She gave it a clean bill of health. There was something about how the dog looked at her when she approached it, some kind of dog-whisperer stuff about the dog not showing aggression or fear. She tried surprising the dog and it reacted in a friendly way. The dog turned out to be a great pet.
It's much like taking a used car to an independent mechanic prior to purchasing it.
So perhaps you can do the same thing - ask the shelter if you can take a potential pet to your choice of vet before committing. Ask about fees for infections, prescriptions, and ask if you'd need to book a vet appointment every time the cat sneezed. Ask the vet if he or she would adopt a cat in that situation.