I'd say if you had any shot at all it would be with CX, so that's where I'd be looking but I agree that your best chance might be 2 x F and 2 x J seats.

I'd try to avoid BA though even just for the surcharges. I recently helped a friend book one-way from LHR-PER via SIN with BA/QF in F then J and it was about $650 AUD with tax and fuel surcharges.