You know, no hotel in New York City offers a group shuttle to an airport. And the outside shuttle companies have all withered away, post Covid. But people still come to visit NYC, and pay $100 or more (with tolls and fees) so for a car service from Newark (EWR.) Or they take a bus or mass transit, to a location with local transit access to the hotel. That's typically "a three-seat ride."
My main reason for posting again is to point out that the train to Civitavecchia stops at other stations besides Termini. You might find a cheaper hotel (I mean, away from Termini) with access to one of those stations. There are express and local services to Civitavecchia, so not every train makes the same stops. Perhaps the hardest thing about taking the train (after handling your own luggage) is buying the ticket at an unattended ticket machine. I mean, it's not hard if you have done this kind of thing before, but it's intimidating the first time, and may require a Chip and Pin credit card. Or, your regular credit card may go through, because the ticket is so incredibly inexpensive. I think our train to Civitavecchia was around $3 in Euros.
Have you been to Europe before? In many cities, the area directly around the train station feels a little sketchy to some visitors. It's often the greyest and dustiest part of downtown. In fairness, I haven't been to Termini in ten years. On that last trip, we stayed at The Inn At The Spanish Steps, and got one free black-car transfer (FCO or Civitavecchia) by booking 3 nights. (I know, it's baked into the room rate!) I don't see that offer on their website today. In fact, they have broadened their business model into something more than just hotel rooms.
I wonder (because I know nothing about it) if you have learned any good potential ways to get from FCO to Civitavecchia, if you go back to that plan. I would expect that it's not easy, and might be an expensive cab ride, because a cab at FCO really wants a fare to midtown Rome. There might be a local bus, but it may stop at an employee locker-room building, and not at the air terminals. I mean, no airport is really set up for non-downtown destinations. So a cab for that short distance could be more than you expect it to be. He probably has to return empty to FCO, and get back in the line to get a big fare to downtown Rome.
As I remember, the cruise company (bus) transfer to FCO and the cruise company transfer to downtown Rome are both over 100 Euros. That's not a free-market price, but it's an ... indication.
I also wonder if the sole good reason for staying in Civitavecchia might be the increased chance to find one of the few car companies that might have permission to take you directly to the ship, out on the pier? But if you stay at a cheap hotel, with a temp-teenager behind the front desk the day your arrive, will that really help? With modern port security, it's kind of unlikely, anyway.
EDIT: Thanks for your comments, sorry if I seemed to be pressing you for a reply, I didn't mean that at all. There is a half-hourly (when we were there) from the Civit.... train station to the Port's shuttle bus circle that I wrote about. I do not believe the port picks up at the train station. Naturally, a train full of travelers is more than will fit on one bus. I'd have some taxi phone numbers printed out in my pocket.