We did this about 2 weeks ago on the Fantasy. i dropped my DH and the kids off at the terminal, drove to Budget to drop off the car and waited less than 10 minutes for the return shuttle. I'm sure I was back to the terminal in less than 45 min all together. Worked perfect!
And they allowed my DH to check all of us in, he took all the documents except for my photo ID. All I had to do once I got into the terminal was show my ID and have my picture taken for the KTTW.
It was not as seamless on the other end however... we waited more than an hour just to get on the shuttle to Budget and waited another hour at the office to get a car. This was partially due to circumstances out of anyone's control however. The ship was about 2.5 hrs late getting in to port due to a death on Castaway Cay the night prior. Because of this the Fantasy was no longer the first ship in, luggage and debarkation were delayed, etc. In hindsight, one of us should have just hopped in a taxi as soon as we exited the terminal and that would have made things a lot faster.