You might consider what you will be doing the day you pick up the car. We spent the day at Epcot with a very late lunch reservation at Via Napoli. After lunch I left via the international gateway while the rest continued touring the world showcase. I took the Friendship cruise all the way to the Dolphin where I was next in line to pick up the car.
If I had picked up the car from the car care center, I think it would have interfered with our day to a much greater extent.
Another option would be to plan a Hollywood studios day. At some point the driver could slip away, take boat to Dolphin, rent car, and drive back to HS parking lot. I bet the whole thing would only take about an hour.