Okay, I've done some research and I'm not confident that any of the waterproofing methods I've found will hold up to repeated laundering.
I'm willing to try a bed "mattress pad" of vinyl or laminated fabric to see how it turns out and whether it seems too hot. My concern with that fabric is whether the vinyl or laminated film will hold up to the ammonia in cat urine, because it will just pool and sit there.