I did a 8 hour wait in the lounge last night. In the F class lounge they have bedrooms and you can use the spa which makes any length of stay really bearable. I think you should negotiate with QR to to give you access to this facility, as it's better than being sent out into DOH in my op. I have also spent 5 hours in the business lounge before, they push two chairs together and insert a table to make a flat bed, then give pillows and blankets out. It's more for the quiet night time period rather than the day though.

Get out and see Doha in the day, sleep on the plane. At least you can add a pin to the map rather than the inside of a Hilton.