We struggled with these same choices when we went to Italy in April, for travel throughout the Basilicata and Puglia regions. We were going to be arriving at FCO on an American Airlines flight, and while I could find flights on other airlines to take us to Bari, the choices weren't great. One was departing about 2 hours after our arrival and I worried about missing that flight if our AA flight got in late. But I didn't want to hang around the airport for 6 hours waiting for the next flight out. We weren't planning to stay in Bari, we were just planning to fly there as a quick way to get across to Puglia. We intended to have a rental car and our plan had been to book one for pick up at the Bari airport.
The plan we ended up with worked very well for us. We nixed the idea of flying to Bari and rented a car at FCO. Leaving the airport was easy driving. Knowing we'd be tired from the missed night of sleep, we didn't plan a big first day of driving. We drove for under an hour, to Castel Gandolfo, where the Pope's summer palace is located. It was a delightful town for a low key day and overnight stay, then we drove the next day to Matera. It was a really beautiful drive. We stopped along the way to see the Abbey of Montecassino. (It was Easter Sunday and we were able to attend Mass there.) For us, six hours of beautiful scenery while driving was a way better plan than sitting in FCO for six hours waiting for a flight!
If you're going to have a rental car in Puglia, you could consider a similar plan.
We took the Rick Steves Best of Sicily tour after Puglia. We returned our rental car at the Bari airport and flew on ITA to Palermo, with a plane change at FCO. Our choices were 50 minutes of layover time at FCO or 2 hours and 50 minutes. By mistake I picked the short layover. I realized my error right away and could have canceled and rebooked but we decided to go for it after I researched how far apart our arrival and departure gates would be. We only had a couple minutes walk from one to the other. If we'd missed the connection, there were many later flights to Palermo on ITA that day. The flight from Bari took off 15 minutes late but we still had ample time to make our connection. Our checked-though luggage made it too! (Our bags were carry-on sized for AA but didn't meet carry-on size restrictions for ITA.)
You, of course, will be changing from an ITA international arrival gate to an ITA domestic departure gate, and will need to go through passport control and security. I'm not sure how much time that transfer will involve but an airport map may help you figure that out. We've flown into and out of FCO a half dozen times and have never experienced any problems navigating that airport. The lines for check in, passport control and security have always been short for us. Others have had different experiences, but we've just been lucky, I guess.