Our last Oct MIA-DOH-AMM was on QR QSuite

The seats were available up to first week of November when I booked ours for last Oct departure.  In fact I already ticketed a MIA-DCA IAD-DOH-AMM just a day before the better MIA-DOH became available, only needed to juggle a couple days but avoided the domestic overnight stay at DCA. 

QR seats come and go - you need to keep a close eye on Flyertalk's AA forum thread QR awards.  There are folks constantly monitor availability - and you get their DPs to act on.

https://www.flyertalk.com/forum/american-airlines-aadvantage/1523205-qatar-airways-qr-redeem-aa-miles-award-master-thd.html

Because canceled QR award seats would usually go back to the inventory,  people often posted on the thread they would cancel their bookings, giving the routing and date - to benefit others who are looking.  It is a community people helping each other.  When I am looking for a specific redemption I would monitor the corresponding thread. 

Pretty sure we will see QR seats come out again.  On top of that QR has MORE seats for its own program which can be linked with your BA program, with instant transfer between the 2 accounts.  In fact sometimes QR would directly take Avios from your BA account if you do not have enough pts in QR account for the redemption.  Of course you will pay YQ on QR or BA redemption, but not outrageously so.