Well, 12 days--let's call it 11 nights--and the fact that you don't want to revisit places, begs the Amalfi peninsula, Naples, and maybe parts of Calabria or even over to Matera/Alberobello in Basilicata.
Thing is, you don't need to spend all 12 days on the A.C. I think a week will be enough, especially given that it will still be plenty busy there in September. You can see Sorrento, Amalfi, Positano, Capri, Salerno, Paestum, Naples, Pompei, Ercolano, and then some. Maybe 4 days base in Sorrento, then 3 days in Salerno.
Problem with going over to visit the caves in Matera and the trulli houses in Alberobello is that it takes time to get there and back, unless you plan to fly into Naples and (as stated above) out of Bari on the east coast--then it might make sense. Otherwise, you can book to fly in & out of Naples, and after Salerno take the train to a place I'm visiting on our next trip--Maratea--I believe 2-3 hours south on the Tyrrhenian coast, for a couple days. Looks beautiful, and relatively unscathed by tourism.
Enjoy your planning!