I think he needs to live within Vatican City because it is a sovereign country apart from Italy. For the sake of maintaining independence from pressure of secular goverment and for personal security, he needs to be in his own country. Otherwise he is a resident of a country which allows a lot of stuff that the church doesn’t. And he is a security risk and headache for a country and city that will have his safety as a distraction from their normal security concerns.
I think occupying a smaller place in the palace, and inviting “roomates” to take up the additional space would be good. Maybe there are a group of third world seminarians who would need a place to stay while in Rome, or something like that.
Technically the ArchBasilica of St. John di Laterano is an exclave of the city-state of the Vatican. Which means, that just as hawaii is an outpost of the USA, St. John the Lateran as well as the other ArchBasilicas in Rome are not in Italy but in the city-state of the Vatican