What might be easier for you is to take one of the direct trains from Florence to FCO. You'd make a couple of stops, one at Termini, but you won't need to change trains and worry about connections and navigating Termini, etc.
The link below brings you to the Slow Europe site and the post is from "GAC" (this poster has contributed greatly to the old Slow Travel site, and also on other travel forums and is generally considered an authority on Italian transportation stuff). The page I'm linking provides info for transportation between FCO and Florence, but you can just use the tips in reverse:
Public Transportation Rome Airport to Florence
Great solution! I wish I had thought of it. The only caveat is that there are only two direct Florence to FCO trains per day. One leaves around 7:30 AM, the other leaves around 11:30 AM. That's it. It's a little over a two hour trip. Arriving at 1:45 PM, for a 7:15 PM flight is a lot of hanging around at the airport time. I think it's better to enjoy Florence some more, or depending on how they use luggage (heavy packers or not), get a taste of Rome. As nice as FCO is, five hours there is too much.