Don't know anything about BA award flights - so will not make any comment
However you can use your AA miles as follows:
- 2 x first class All Partner Award to NZ - flying QF to SYD or MEL (A380 F hopefully) and on to NZ. Cost is 45,000 miles each = total 90,000
- 2 x business class All Partner Award from NZ back to SIN - perhaps flying CX via HKG. Cost is 35,000 miles each = total 70,000
Grand Total 160,000 miles. So you are 10,000 short. It will cost you about USD300 to buy 10,000 miles.
See the All Partner Award chart
here
Notes:
- for the same cost in miles you could travel as far as Tahiti instead of NZ
- no stopovers allowed
- no fuel surcharges, just the local taxes (maybe about AUD150) plus a booking fee of USD25
- availability is likely to be a problem - especially for QF F on an A380