Am likely to be flying this often in the next 6 months or so.

Ideally would like to do this with one stopover but seems unlikely (whether OW or not).

I think I can make it LAX-DFW-LGW-BRI with two stopovers, but the LGW-BRI seems seasonal - is that right?

Is BA the only OW airlines which flies to BRI? And is so only from LHR? (I have only looked in the OW timetable).

As I understand it, Bari is primarily a holiday destination, so yes - the BA flights are seasonal, and they go only from LGW (not LHR)

But it is only 240 miles or so from Rome, so at the speed Italians drive that won't take very long - maybe best to look at flying in there (via LHR) and then going by land?

Driving is an option, but do remember this is southern Italy where things are considerably more chaotic than in the north. I once rented a car in Naples and almost caused an accident by braking for a red light. Also beware that the highways can be narrower and less accomodating to truly high speeds than the ones in the north (especially tricky are the 30-feet merge lanes to get onto the highway). I'm not sure I'd recommend a 240-mile trip by car after such a time in the air. It would certainly be fun, but could take longer than you bargain for.

Your best OW option (other than the LGW flight) is to fly to FCO, then take one of the many AZ flights to Bari. If you want to do *A, go via MUC and take the LH flight down. Other options would be to go via LIN, MXP, VER or VCE.