That route has very limited availability for CX MPC members, much less *other* airline partners. Most of the time, CX will only release ONE seat per flight, and after that is booked, *MAY* release another.
But JULY is getting into peak season, and thus CX may be unwilling to release any more award seats since they are able to sell that seat as full revenue.
Having just an asiamiles account isn't going to help you especially if you don't have miles there to redeem. (Asiamiles and MPC members have the added ability to waitlist on CX awards).
Your best bet is to keep contacting BA and ask them to check for availability.