What about someplace else in the Middle East?

You could do JFK-oAMM-xMAD-xLHR-oMCT-xLHR-SIN or substitute DXB for MCT. If you wanted just to do the LHR-MCT turnaround (really not a bad experience) you could add a day in AMM, since LHR-MCT departs in the morning, and MCT-LHR leaves at midnight. That would give you 5 hours in the T5 arrivals lounge to shower and nap before heading to T3 for the Qantas flight.

(AMM-xMAD-LHR on 19 October, LHR-xMCT-LHR on 20 Oct. returning 0645 on the 21st, then LHR-SIN.)