HI, This summer we went on the Constellation. We flew into Gatwick. I had transfers through Celebrity, but it was going to be a 3 hour wait for the bus! My hubby rented a car and it took 45 minutes to get to the ship.
The car rental place in Dover is almost within walking distance to the ship (almost). We had a free drop off at dover and the cab was waiting for us at the car rental place there. I would suggest you rent a Van.
By the way, if you don't ask for it, you will get a manual trans, not auto. Fortunately, hubby knows how to drive a manual!