I rewrote your itinerary to have a better overview:

ARN-xLHR-YUL-JFK//LAX-LIM-EZE-SCL-IPC-PPT//AKL-SYD-DRW-PER-SIN-NRT-HKG-BOM-xLHR-ARN

If I counted correctly, you only have 19 segments (including surface transfers) in your itinerary so you might add one somewhere.

Another FFP option may be Alaska Airlines (AS). They give you 100% on AA, QF, LA, 50% on CX, 25% on BA; unfortunately no credit for JL. If you change your direct SIN-NRT on JL to a SIN-xHKG-NRT on CX (using your additional segment here), you will get miles. On top of that, AS partners also with Delta (DL), Air France (AF) and a few others, so it might be very attractive an option.

From my calculations, you would come up with following total of points (approximates):
- AA: 18400
- QF: 26000
- AS: 24000 (without changing routing)

Therefore I’d go for AS or QF.