I think we returned a car to the Hertz rental at the airport several years ago before we flew home. Although I had been warned several times and had specific instructions we drove around the actual building 3 times before we found the obscure entrance to the building and the drive in was kind of tight and space for parking very tight. Very hard to go around the car and photo record that there was no damage. I'd opt for the returns away from the airport where it won't be so hectic on return.
Just to clarify
... you want to drop the car off at FCO
.... you then want to get to accommodation near the Vatican.
.... you want later to get back to the airport.
Is that correct?
As Bonjours says you can drop off the car in Via Sardegna. But that’s not near the Vatican and does involve some risks for the novice of entering a ZTL street.
This may help you get into rental return at FCO, including the human experience of getting it wrong and going round again.
Then get across to airport arrivals and the taxi rank. 48 euros to your destination. If it is inside the Aurelian Walls, check with your host.
Same fixed fee to return to the airport.