Tia, the reward flights on this route go really early, I started scanning 355 days in advance.
However, have you tried looking at partner airlines? Im not an expert on the upgrade using avios conditions, as I never seem to have any luck ( plus it always seems to be work out very expensive for us) but assuming you arent also using an amex 241 then it could be worth ticking the box on BA.com and looking at the partners, Qantas for example,thereby getting something booked with avios . It at least gets you booked in business. Then looking each morning to see if BA metal comes up.
This worked for us before when I used avios for Qantas business then managed to switch to BA First for fewer miles when availability opened up, deploying our 241.
Just looking at the 11 Feb now, there are business class reward flights on Qatar , also indirect flights with BA to DXB though I expect the usefulness of my advice depends on what your specific needs are and how many avios you hve to play with.
Oh yes, also worth thinking about looking at Abu Dhabi AUH and getting a car transfer, availability always seemed better than DXB when I looked at this. ( ah yes , BA.com is showing CW reward availability to AUH on 11 Feb returning on the 16th) Its 90mins transfer , less than that if you are staying south of Dubai. Taxis are cheap.
Hope that helps, Ive spent hours for our own flights because of the same challenge.