Qantas codeshare flights operated by Emirates (with a QF flight number) between Asia and Australia/New Zealand are not eligible for our Chauffeur-drive service. If a customer would like to request Chauffeur-drive for flights between Asia and Australia/New Zealand, they must be booked on an Emirates operated flight (with an EK flight number).
The OP is eligible for the chauffeur drive in SYD and given this is an outstation for EK they should be able to arrange for it.
I have seen this happen for a SYD-BNE-SIN itinerary, as well as an ADL-MEL-SIN itinerary.