If you're not able to get something that opens up when the schedule does, one of the best things that you can do is set alerts and check every single day. I'd also check daily for flights via DOH... Qatar sometimes releases a chunk of seats all at once, and they can be booked with Avios...

Don't be concerned about booking the roundtrip at once - do it as a separate outbound and return.

If you see even partial routings that will work, consider booking something like a USA-DOH and then booking a leg in economy (and keeping an eye out for J seats that you can rebook into later).

I'd also see if any BA Premium Economy fares look reasonable and if you can get into J on the overnight legs at booking, it could be.a reasonable use of Avios. But beware of the tax differences ex-USA.