With 4 pc of luggage it's definitely a cab. I see you and your partner walking from your hotel to Termini and then to the platform that the Leonardo Express goes from, OMG... Then walking from the station at FCO to the CI counters, OMG... As iapetus suggested, the cab will bring you straight to the building, and you can always ask your hotel, when booking the taxi, to ask for a larger car, e.g. a Fiat Multipla or a Renault Espace or Scenic, or something like that. This should not be a problem.

The rest of my post will be sort of out-of-topic I assume that you're on vacation and not flying home from a year-long stay abroad. Why on earth do you carry so much stuff with you, needing 4 pieces of luggage? When travelling with my wife, we usually go with 1-2 pieces maximum and we manage easily. Our transfers are much easier, and we could even go cheaper such as even using LCCs with lots of restrictions. Try it out, your quality of life will improve tremendously.