You could spend the entire time in Sicily and face very painful itinerary cuts. Early September would probably still be quite hot there. I do love Sicily, but I'd recommend a different month.
In addition to a bunch of time in Rome, I spent seven nights in Naples and four nights in Salerno in Feb/Mar 2023. I really liked both and could have used more time in the area. Lodging in those cities is usually a lot less expensive than on the Amalfi Coast or in Sorrento. You may encounter some unpleasantly hot days there, as well. Be sure your lodgings have reliable air conditioning, and keep in mind that if you day trip away from your base city, you won't be able to pop back to your hotel easily for a 30-minute break. Early starts are advised.
If you end up spending a good bit of time in the area around Naples, check out the Campania Arte Card. The one-year card was very worthwhile for me, but there's also a shorter-duration card, I believe. The Campania Arte Card covers Pompeii, Herculaneum, Paestum and other excavation sites as well as the zroyal Palace at Caserta and a lot of things in Naples. It was great not to have to rush to cram everything into just a few days.
It's very difficult if not impossible to find a place to buy the Campania Arte Card in person, but it's very easy to do so within the Arte Card app. It's very important to make a record of the numbers that show up during the purchase process. One of those numbers will get you into the covered sights if the app fails to open for you (which happened to me twice).
