It turns out that you can fly BA in Business JFK-LHR-MAA on the 5th, and QR in Economy MAA-DOH-JFK on the 27th with Avios. But if you want to get the special multi-carrier redemption pricing, you would have to tack on a short segment on AA for some date after your return to NYC, so that you have at least two BA partners on the award. Or you could book MAA-DOH-BOS in Economy on QR on the 27th, and then tack on an AA BOS-JFK segment that night, or on a later date. Again, if you fly the outbound in Business, but return on QR on Economy via DOH, the whole trip should price at just 200,000 Avios for two, (plus fees and taxes of ~$1,650, total), since DOH-JFK/BOS would be the longest segment on the itinerary.