A common recommendation here is to stay in Ft Lauderdale and take one of the frequently recommended shuttles to the port the day of sailing.
Have you contacted any of the frequently recommended shuttle services mentioned here to see what they offer in your situation? SAS, QLS, Jiffy Jeff, GO Shuttle and a couple others.
Also a common recommendation is Larry's Limo, which, for a party of 4, wouldn't be too expensive.