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As someone who has often had fosters who won't take a dog into their home unless it was cat-tested at a shelter, I hope the solution isn't "we don't ever do it." When I was at shelters that had outdoor managed feral cats in the parking lot, those were great helpers for me. The big old Tom cats would sit and stare at the dog from a distance, and we could evaluate the dog's reaction on-leash, far enough away that the dog couldn't bother them. There was also a time when one municipal shelter had a dog-loving "office cat" who loved to interract with dogs--he was a great reader of dog body language and fantastic at helping out (and he went home with one of the directors on weekends and holidays, I think).
I know several rescues that have passed up pulling several dogs from high-kill shelters that desperately needed those rescue pulls but lost out on them because the shelter "had a policy" of refusing to allow cat testing...and the rescues had foster homes that wouldn't take them without that because they owned cats. Those dogs were almost certainly euthanized because of that "policy." That sucks, but if a foster home won't take a dog without a cat test, then the rescues can't work with shelters that say "no." If you're in an area where every overcrowded shelter is competing for rescue pulls, some rescues will just move on to other shelters that are easier to work with.
Try to find a way to say "yes" in a creative way that minimizes stress for the cats -- unless you're in an area where rescue pulls don't matter, and then do whatever feels comfortable.