Is there any reasonable way to fit Bari into a Venice, Rome, Cinque Terre trip?
Depends on how many days you have and how you want to travel (fly, drive, train). If the above is the order of your itinerary I would suggest Venice, Bari, Rome CT. It's 8 hours by train from Venice to Bari (changing at Bologna). Then from Bari to Rome is 4 hours. Or you can fly via Rome in 3 hours. In which case you might want to readjust your itinerary a bit.
While Trenitalia sells tickets, we prefer to find and print most train schedules with DB. For flight comparisons, try www.skyscanner.com.