I'm getting ready to book a flight from Boston to Brindisi on April 1, 2022, return 2 weeks later, probably Bari. The new ITA website is non-functional. Do you have any suggestions for airlines? I though perhaps Swissair.
Sorry no suggestions, but I will say Bari is a nice airport when hiring a car, as the roads around the airport are extremely quiet, making it easy to get used to the car / driving conditions. Assuming you'll be heading south on the Autostrada, the traffic does get a little busier around Bari, but that's only for 2-3 junctions.
If returning there at the end of the trip, I'd recommend Trani as a useful place to spend a couple of hours, if you worry about the timings of a longer journey to an airport. It's quite charming and close to the airport, so makes it a very useful de-risk option that still rewards.
You can search Google Flights. The search I've just done with your parameters would indeed show a good deal on Swiss, although the return connecting in both Zurich and Montreal would be a long day. With a date adjustment, a return on Lufthansa via Munich looks good.