Can we count on a very early morning cab to Rome airport?

The night before you are headed to the airport, ask your hotel to call the cab for you. Give them the time you want to be picked up, ask them if it's reliable. I would assume it is.

I would NOT try to flag down a cab on the street to get to the airport for your flight home. Been there, done that, never again: In Warsaw, wandering the streets in the early morning, dragging our bags, seeing no cabs where they had been plentiful the night before, looking for a cab and starting to get worried. First cabbie I managed to wrangle wanted €150 for the ride to the airport. I laughed, he said "oh come on, how about €75?" (We ended up paying about €15 to a different guy). Only reason we had to go through that was we were at an airbnb, so no hotel services. In that case, I'd still arrange the cab the night before (but have a backup plan/cab phone number and working phone). No last-minute drama needed.

You can book ahead of time with “Rome Cabs”. They will pick you up at your hotel at specified time.

We’ve taken cabs to the airport for much earlier flights than yours. Our hotel arranged for the cab and they were reliable and zipped through the empty streets at our ungodly hour.

Two years ago we stayed for a week in an apartment in he Trastevere neighborhood. We knew exactly where the taxi stand was by the Sisto Bridge. We were confident that there would be a taxi waiting at all hours so we used the taxi stand.

If you are staying at a hotel let them prebook for you. If you are in apartment on a street accessible to taxis you can prebook for yourselves or walk to a major, known taxi stand.

We have done it a number of times - both shared shuttle and a taxi --- with no problems. We have had a pickup as earily as 4.30 (Never again). Always let the hotel arrange it. Generally the taxi is early. We would rather spend an evening in Rome than the night at the airport.

The idea of flagging down roaming taxis is an American thing, maybe a big city east coast thing. We've found that hotels in Western Europe have established relationships with taxi companies or even specific drivers, and know who's reliable.

Second, Stan -- that is our experience also. You don't flag down cabs on the streets of Rome. You need to use a taxi stand.

Hailing taxis is in Italy is not done. You either call them or walk to a taxi stand.
Taxis are reliable 24/7.
There are several cooperatives (Italian taxis are owned by the taxi drivers, as the law prohibit giving licenses to companies) but the taxi owners are generally partners in cooperatives to share services (dispatch, marketing).
This is the largest taxi coop in Rome:
https://www.3570.it/services
Romecabs is a limo company that is reputable. Maybe slightly more expensive than a taxi, but good service:
https://www.romecabs.com/

There's been a lot of great advice given here. I'd ask what your decision points are? On that day, are you traveling into Termini and it would be easy to progress to the airport? Are you thinking you want a "last supper" and a last nighttime stroll to Trevi Fountain? I completely understand that. Do you want a good sleep before boarding a long flight? Personally, I like to build less stress in my life but I do have an "I can always sleep on the plane" attitude. I'd either stay near Termini Station and then jump on the Leonardo Express in the morning or stay at the airport. Maybe another neighborhood and train would suit your needs. If I needed a taxi, I'd prearrange it. Getting to a flight on time is always something I like to plan out but that's me.

We have used www.suntransfers.com many times and havent had any issues. They will pick you up from your hotel in Rome and get you to your flight no matter how early. We have had 3:00am pick up times before.

Call the cab service the day before you depart and schedule a pickup from wherever you are staying. The driver will be there punctually and without fail at the agreed time. Roberta D has provided the cab service names you may contact.
Don't feel like you need to bargain hunt. The rates are all the same and will be precisely what you are quoted. This will not be an NYC cab ride where they take you around in circles trying to rack up the meter. The driver will know the best route and will get you to the airport without any craziness...
We usually stay in and Ab&B, last time in Trastevere. I have no clue how the cabby found the place where we were staying but at 4:00 AM, there he was. We always go out for a great last meal and a walk on our last night....

Many times, I've arranged in advance for transportation to the airport. Most recently (early January), my pick up for FCO was at 3 a.m. for a 6 a.m. flight. No problems.

Having transportation arranged by the hotel may be a good solution, but not always. It might be a shuttle of some type, a commercial car service, or the desk clerk's brother-in-law. We got an inlaw once but it turned out ok. Have you thought about using a private car service? It will cost a bit more than a taxi, but not much more. The service we've used very early in the morning (and will use again) is Rome Cabs. It is definitely a car service and not a taxi company despite the name. We just send an email detailing our needs and get an answer within 24 hours. This is the link to their site. https://www.romecabs.com/.

We stayed in the airport at Hellosky. The rooms are very neat and clean. It was nice to just walk out of the room in the morning right into the airport.