There is no car rental at the cruise terminal (2) where
DCL docks, the closest is Enterprise that's a ten minute walk. Yes for Eastern docks with cruise luggage you need to get a taxi at the cruise terminal. It will be a ten minutes drive. It will cost up to £10. There are loads of taxis but often there are queues there, space is limited.
If you click the link of my trip report below, To Russia with Mickey bars, there are loads of photos, maps, travel information for Dover, and 12 night navigators from this year.
As well as my cruise embarking and disembarking in Dover, I went to the previous Magic call at Dover, and watched embarkation and disembarkation there, as well as undertaking it.
Plan that the ship may come in later than other ports such as Barcelona and Port Canaveral, we were in at 6.10 and as the terminal is small, disembarkation was by deck and tag and luggage not so organised, some were getting off at 10.30 am.
Taxis have queues and even pre booked ones are held in stacks outside the port.
I think you are on our cruise we have very active cruise meets groups. Do join.