Airport security check in at Da Vinci Rome

Each airline sets its standards for check-in time limits.
I presume you are flying Alitalia (the other option would be Ryanair).

Alitalia, for EU Schengen flights and within Italy, requires that you have completed your bag check-in and have a boarding pass in your hands at least 40 minutes prior to departure.
I can't possibly imagine that the check in line would be longer than 30 min for a flight to BRI, therefore being at the airport 1h15 min prior to departure would be, IMO, more than sufficient.

https://www.alitalia.com/it_it/volare-alitalia/check-in/tempi-limite-check-in.html

You don't describe your full travel plans. If you arrive from an international flight and go to BRI right away, it might make sense to fly. But if you are already in Rome for a few days, it's actually just as fast (and more comfortable) to go by high speed train from the Roma Termini station.

Check train schedule and prices below:
www.trenitalia.com

Thanks for the reply. We are weighing whether to fly, take the train, or drive. For this thread I'm just collecting data on the flights. I'll post additional questions regarding trains and driving. We had originally thought we could travel from San Gimignano to Bari in a single day, however, that now seems impractical. It appears that you can't comfortably get from San Gimignano to Bari in a single day, no matter if you go by plane, train or car.

So, if I understand your post, it sounds like Da Vinci security for domestic flights could be as little as half what it can be for international flights.

Maybe fly from Florence or Pisa? I think that would be a lot faster than finding your way to Rome and flying from there.

The only direct flights to Bari are from Rome and most from Pisa and Florence end up going to Rome first. By the time you add on the connection at Da Vinci, it would be faster to take the train. If we can't get a direct flight, there's really no point. The flights out of Pisa, Florence and Perugia often have some weird and lengthy connections - Munich, Zurich, Bucharest, Tirana (Albania) and even London. We're trying to minimize our travel time, so if we fly, we'll do so from Rome.

I've driven from Puglia (Taranto, Bari, etc.) to Florence and back more than once all in a day. It takes less time than driving from San Francisco to San Diego, which I have also done in a day and going to do twice this coming Labor Day weekend. So I don't know why it would be impractical. A long drive for sure, but lots of Americans are used to driving these distances. It's about 7 hours plus pit stops from San Gimignano to Bari. However, given the availability of high speed trains in Italy, that would definitely be my choice in Italy. That is not an option I have in California where Amtrak takes 14 hours from Oakland to Los Angeles, 5 times as long as Florence to Naples, which are just as distant.

Ryanair for sure flies from Pisa to Bari (BRI).....and Brindisi (BDS) too.

Ryanair flies to Bari from Pisa and Perugia, but the times are limited - usually only once/day - and the times rarely seem to be compatible with our schedule. And besides, it's Ryanair.