I haven't used FLL in at least a couple of years, but last I was there I don't recall anything about any airside inter-terminal transit option for passengers to go from Terminal 1 to 4 and thought it would be like other inter-terminal transits there: going landside and then dealing with another airport security screening experience after the bus or walk.

Does FLL still have the issue where TSA denies passengers from going (and perhaps even going back) airside more than 2 hours before scheduled departure on some cruise-heavy days? It may be something to consider to be an issue if you have a transit time that is more than 2 hours.