Upvote:4
On the web site of the US diplomatic mission to Italy, you can read about processing times for passport applications:
Processing Time
Except in emergencies, U.S. passports are now produced at facilities in the United States. These passports will be ready for collection approximately three weeks after submission. For lost or stolen passports, a limited validity passport can be issued the same day the application is made if the application is received before 12:00PM. Otherwise, it will be processed on the next business day.
In combination with a comment noting that a recent passport renewal in France took less than a week, you can conclude that the three-week timeframe is unlikely to be exceeded. You can probably therefore use the normal renewal process and still receive your passport in time for your trip.