Hi, we did the reverse of what you want to do. We took a very short cab ride to a Rental Car Agency right outside one of the Port Gates, as far as a free shuttle pick up or drop off we did not enquire as I was pressed for time so they may have Shuttle service available. The rates were pretty reasonable at the time (April 2013). I would imagine that you could pick up the car at airport & then drop it off at the Port Location.
We went to Thrifty/Dollar Rental Agency at 2400 South Miami Road, Fort Lauderdale, FL 33316
Phone: (954) 463-2958
Returning was not a huge problem as we rented the car & dropped it off later in the week at Miami Intl.
We also stayed near the Sawgrass Mills Mall but usually stay at the Dolphin Mall in Miami, we wanted to try a different location. After doing so our own experience was that we liked the Dolphin Mall far better in Miami. We felt that the Dolphin Mall was much easier to navigate, stores, entertainment, food choices (besides food court) were great as well as the Hotel is literally in the Mall Parking Lot. We just rolled out of bed & rolled right into the Mall pretty much LOL.
Hope this helps a little, enjoy, have fun, & most of all be safe.