In theory that sounds like a great idea, but due to the distance involved, it most likely isn't possible unless you had only 2 pilots flying to meet midway. And I think (without determining the miles involved) that you will need more than 2. At some point in the transport, it might be possible for 2 pilots to meet, swap cats & return to their airports, but the beginning pilots would be long gone from their drop off points. Does this make sense?
If it were me, I'd post separate transports for the cats & see if they can overlap once you get some responses.