I'm not an expert on manipulating awards bookings so can't give any advice there. But as a BA Silver, I can't get any availability at all in July from London to Athens.

Athens is well served from London by both easyJet and BA, which are offering return prices on random dates in July for between £120 and £140 including taxes. That's a good price for that length of trip, but won't last forever.

Given they're flying in with QF and the BA flights depart from LHR, that might be your best choice. easyJet fly from LGW and Luton, but may be less understanding over luggage, delays etc. than a oneworld partner.