I believe the fixed fare only apply to certain condition, and certainly not at night (at the very least there would be a further premium).

The problem is that the last train from FCO to Termini is at around 00:23 and the next after that is at 5:38 in the morning, so Taxi is the only option. Certain Car Rental companies may run a self service kiosk out of hours where you can simply pick the key and drive away (I suggest a full video, 360 degree, around the car - including the roof to show the presence or less of the radio arial - and pictures of any existing damage). And other than Rental, the Taxi is the best option. There are two fast trains options which are quite comfortable, Freccia Rossa and Italo that only takes 1hr 20 - 1hr 30 circa from Termini to Florence, and you will be right in the middle of the city (you could easily walk all the way to Duomo).