The distance from the port of Miami to FLL is 24 miles and about a 30 minute drive once you are on the road. Customs at the port of Miami is not always fast, plus problems can happen like they did last year when we were leaving the Liberty. Customs had to shut down for technical problems for over 30 minutes. Nobody could leave.
Even assuming you did the self-debarkation, which if all goes perfect can start at 6:30am, you have the Customs checkpoint, getting out of the terminal, and then getting to your ride to contemplate. Now the hassle of getting out of the port in the rush of traffic. That will eat up enough time to make the choice of an 8:30am flight a bad idea.
FLL isn't as user-friendly as it used to be, according to my sister who leaves nearby. She now uses West Palm Beach for flights when at all possible. The lines can be long at TSA.