Yeap I agree with Happy comment, most likely all your trip will earn miles since they are in booked in business class. So narrow it down to three CX, QF and AA and check which one you would earn the most with and maybe can maintain the status with easier and validity of your miles.
What an option is let one of your party sign up for the best program and the other for the second best, That way you have two different ff program in the same alliance. Unless you have plans to combine the ff miles together.
What is also important which program has more awards available and lesser tax.

Good luck and it looks like an amazing Itinerary

Happy travels!