I've flown cats as cargo on passenger airlines, it works just fine. American Airlines has good advice on shipping cats.
If I was in your situation, I'd try that, because driving cats that far with a litter box in the car does not sound fun. Maybe you could find someone, a cat boarding business perhaps, that'd board the cats and transport them to or from the airport when needed. I really think that'd be a lot better solution for everyone than driving them cross country.