Just tested and was able to book an infant without seat in 1st class on Qantas as a paid booking, so don't think that it is class of service
Looking at AA's fares, they do not allow travel to Australia on QF, only AA
QF's fares do not allow ticketing by AA
Since there are no fares that AA can sell , the only option will be if QF is prepared to issue a ticket
I would suggest calling, explain that AA cannot issue a ticket due to QF's ticketing restrictions and see whether QF will sell an infant standalone linked to the AA issued SYD-LAX flight - just for the SYD-LAX flight
if you can get QF to agree to issue a ticket, you will be fine - if not, then you ,may need to rethink how to tixket an infant