The difficulty might be booking the BA codeshare on QF from SIN to PER. I suspect the only way might be to ring BA and ask.
EF turns up no results with this booking. The only fares for BA on the SIN PER leg oneway are H,B and Y.
HEEOWSG BA H One-Way 371.00(AUD)
HEEOWSG QF H One-Way 399.00(AUD) E
BEEOWSG BA B One-Way 504.00(AUD) E
BEEOWSG QF B One-Way 531.00(AUD) E

If you're after EQPs, then the B fare is not too much more.

Just did the following dummy booking on ba.com

Check your flight itinerary details
Outbound - Wednesday 16 Jan, 2013
SGD392.20

Email price quote

Currency calculator
Change flights
19:45
16 Jan 01:05
17 Jan Changi (Singapore) Perth BA7378
Qantas Airways Ltd Economy
Price breakdown
Passenger Fare Taxes, fees and carrier charges per person Total price**
1 Adult SGD163.00 SGD229.20 SGD392.20


Perhaps you might find a kind agent to make a constructed fare (SYD-SIN + SIN-PER) for you.