You can hire a towncar service, or rent a car. You may want to look into renting a car at the Dolphin...it's not too far from the CBR. I'm not sure if there are any Mears shuttles that run out to KSC.
I think you should rent a car. It's a good hour drive from Orlando to the KSC. Of course, it took me 1hour 10minutes for Saratoga Springs, but I probably didn't go the most direct route.
KSC is really a good side trip. Leave yourself lots of time. In other words, don't get there any later than 10am (try for earlier). I think it stays open until 6pm. There is really a lot to see there and the IMAX films are great.
Rent a car. Alamo will send a shuttle to pick you up at the Carribean Beach then bring you over to the car care center near the Magic Kingdom where you can pick up a car. It is probably the least expensive way to go.