For a 6AM flight you probably don't need to be there 3 hours ahead - it's probably a schengen flight not to the US/Canada? And what time does security open at FCO? Smaller airports don't always open lines until 1-2 hours before the first flight.
So I'd get there at 4. Personally, I don't like moving hotels so I'd just keep my last night in my hotel in Rome and get a cab. Arrange it the day before or earlier, it would come at 3:30, so I'd be up at 3. No fun, but I find that better than packing my stuff and getting it out of the room by noon (checkout time), not having a room to take a siesta or clean up in the afternoon, coming back to get my luggage that was held at the desk (because I'd still be out sightseeing past check out time), then checking in to a new hotel for what amounts to half a night of sleep. Also, I like to take an evening stroll around the city for my last night in town. Hard to do when you are staying by the airport!