I do this a lot for work and we have the same policy. Best bet (from experience) if you want to stick with BA is to go BA11 to SIN then switch to BA15 and the same in reverse (BA16 to BA12). Then sign up for alerts on Expert flyer for J reward seats on those flights. The only one you really need it for is the LHR - SIN leg and back the other way. On the way out the SIN - SYD flight is so short that you won't sleep properly anyway even in F and on the way back SYD - SIN is a day time flight where you don't really want to sleep to set you up for the SIN - LHR.