What is the best method to get to Bari from FCO?

If you go by train the luggage limits do not apply. However if the train leaves from Roma Termini at 9:30, it is risky to book it ahead of time with a flight arriving at 7:45. The disembarking and passport control procedures alone, and the walking to the Airport station from the terminal will take you at least one hour. Then you need to take the Leonardo Express train from the Fiumicino Airport station to Termini (up to 30 min wait plus a 32 min ride), then you need to walk to the platform where the Bari-bound train will depart from (5-10 min approx).
If you go by train upon landing you should buy the train ticket on the spot when you are at the Fiumicino airport station. You'll have to pay full price, since discounted tickets will not be available on the spot, however discounted ticket have restrictions, and if your flight is late and you miss the train you book, you lose it all. There is no risk of having fully sold out trains. There are many trains to Bari throughout the day, and they don't sell out.

Regarding flights currently ITA Airways has 4 flights from FCO to BRI:
9:30
13:30
17:25
21:50

RyanAir has the following 2 flights:
7:55
19:35

RyanAir has strict restrictions on carry on size. You can find their requirements on their website.

You’ll already be at the airport. So a one-hour flight beats 5 hours on the not-real-scenic train route, if the timing works out.
www.Skyscanner.com

We flew, cheap and easy!

Do you think we would be safe booking a 9:30 AM Flight? can’t find another one until midday and when I look at sky scanner for a round-trip ticket, the 930 flight disappears.

9:30 is Really, really, really, really risky. You have zero room for a delayed arrival or a backup at passport control.

I think 9:30 am also is really risky. Unless you pack really light, you are going to have to check your luggage on ITA. A regular U.S. carryon is not allowed.

I would either take the 1:30 pm flight or take the train and not buy tickets ahead of time. That way you can take the next train without having to build in a buffer.

There is a fast Frecciarossa departing from Roma Termini at 10:00, which would get you to Bari Centrale at 14:10.

There are Leonardo Express trains departing from the airport at 8:23, 8:38, 8:53, 9:08 which would allow you to make that train. That would be faster than any flight departing after 11am.

The next fast train is at 13:05 from Termini (arr. at 17:14)

I would shoot to make the 10:00 train. Even if they sell out the Standard class (unlikely), I would pay extra for the premium class or even the business class (they never run out of those seats). That would still be cheaper than flying with ITA airways. The full price to Bari Centrale is about 75€, and an extra 10€ for premium class.

Fly. It's a no-brainer.

For our upcoming fall trip to Puglia, I also booked my flight with points, and similarly could only get as far as FCO for a reasonable cost in points. No problem, there are plenty of flight options to choose from throughout the day for the last leg from FCO to BRI.

Our inbound from USA is scheduled to arrive at FCO at 8:15 am. I picked the 1:30 pm departure from FCO to BRI on ITA. Cheap, easy, direct, no worries. Four and a half hours to connect should be plenty of time even if our inbound flight is a bit late; in a worst-case scenario, if we are spectacularly late in arriving at FCO, there are later flights the same day we could change to. We're flying business class from the US to Rome, so I'm confident I'll get good sleep on the flight over and will arrive well-rested and smiling. Killing 4-5 hours in FCO is trivial - we need to buy SIMs for our phones, we'll do that in the airport to burn an hour, otherwise we'll just have a few hours to relax in a lounge.

Our ITA flight to Bari arrives 2:30 pm, which is perfect for us. We will be in no great hurry, should have plenty of time to get our rental car upon arrival, find our way into our BnB in the old city, park the car, go out for our traditional orientation walk around the city that afternoon, and then just reeelax over a lovely meal before crashing for the night.

Sure, included luggage is often very limited on intra-Europe flights. That's easily solved by paying for an extra bag if you want/need to; we did that.

This should be easy and cheap. Just allow enough time to be sure you won''t miss your onward flight, since you will be on two separate tickets. Worst case: FCO-BRI is a pretty a cheap flight, even if you had to buy a same-day last-minute ticket in a pinch (and, as you can see, you also have train options if needed). This should be easy peasey, take this off your list of things to worry about and enjoy your trip.

Every flight and connection is different, but for what it's worth we flew straight from FCO to Brindisi (not Bari) in 2018 with a 3.5 hour layover and even though our flight from the States was on time, we needed every minute of that layover to make the Puglia flight.