Kristina,
It's possible that someone on the boards knows of a better plan, but have you considered staying the night before in a hotel close to the airport? There are several, and it might give you considerable piece of mind the night before your horribly early flight. I've never done it in Rome, but I have stayed the night before near the airport in Paris. Sad not to have "our last night..." in Rome, but infinitely less stressful.
Alternatively, there are private car services that advertise 24/7 rates, so that might be a good choice. You might contact your hotel's concierge and ask if he/she can make a suggestion. You are surely not the first guest to ask!
Best wishes!
Debbie