You mention both Q class and K class in your posts, and depending on which of these you are booked in could be a deciding factor as to which program to credit to.

BA Executive Club will only award 25% of miles per mile flown on either Q or K class with Qantas, so it's a very poor earn ratio there. There's also no chance of earning status with BA in these booking classes, although you haven't indicated that this is a priority for you.

Qantas Frequent Flyer, will award 100% mileage on all of your QF flights which are over 1,000 miles in distance. There's a minimum miles earn of 1,000 for QF flights under 1,000 miles distance. You will also earn status credits on these fares should you wish to build towards elite status.

If your flights were booked in K class, then I'd suggest you join the AAdvantage scheme, as you'll earn 100% mileage and they'll help count towards status. Being based in DC and occasionally flying AA for work may provide you with more opportunities with AAdvantage.

As I said earlier though, it all depends on whether you're booked in Q or K class. If it's Q class then I'd go with Qantas FF, if it's K class, go with AAdvantage.

I hope this helps, and have a fantastic trip to Oz and NZ.