I've flown American from FCO many times with carry-on luggage, online check-in and a pre-printed ticket. If you were familiar with FCO, you could probably get away with 2.5 hours. Since you're not, I suggest you stick with the recommended 3 hours. The security lines are unpredictable and the walk from the terminal to the departure gate is now very long. I never stop at the counter unless I have a question, and putting a security sticker on the passport is no longer done. I suggest you add the American Airlines app to your cell phone. It can be used to check in and to monitor updates on your flight.
Edit: A couple of ideas to allow for a little bit of extra sleep. Unless your hotel is within a very short walk of Termini, take a taxi or shuttle directly from your hotel to the airport. If you're quite close, there's a Leonardo Express departing around 6am that will work for you. Skip breakfast at the hotel (unless they give you a take away) and have breakfast once you're past security and passport control at the airport. There will be multiple options near the gate and along the way or, if your hotel gave you a take away breakfast, you can buy coffee to go along with it.