Because you can pretty much interchangeably credit AA and BA to each other's FFP, you can fly both airlines and decide which quality you prefer. Keep in mind BA seat selection charges, though, including both W and J, will apply for your son if traveling without you until he reaches enough status to get them waived..

That said, the best routing looks to be MAF-DFW-DUB, all on AA metal. You'd be doing this trip regularly enough that removing the uncertainty (and worse timings) of an extra transit in London seems pretty desirable, especially during strike season. While this is a Pre-clearance route, I'd still get Global Entry for both of you if you don't already have it. It's of limited utility in DUB (you end up in the same holding pen as everyone without it, you just spend less time in line) but will become more useful if rerouted. It will also include Pre-Check for your outbound flights.

BA appears to sell the AA DFW-DUB flight with a BA code. If you decide to credit to BA, these segments should count to your minimum eligible flight requirements (if booked with BA flight numbers).