Have you looked at Qantas

QF codeshares with EK on services so can cut journey connections down by doing GLA-DXB-SYD/MEL or GLA-DXB-SIN-MEL/SYD

It is from GLA rather than EDI but the 2 are not very far apart and can save on connections

If booked on the QF flight numbers will earn miles etc to AA and also with a QF flight number can undertake the Platinum challenge

Putting 28 dec in for departure from GLA and a cpl of nights in Singapore, ITA is pricing at GBP1433 for a fare outbound in M return in L which is 50% mile earning and 100% of miles earned for EQPS

GLA-DXB-SIN-SYD-DXB-GLA is 22288 miles , so easily meets the 10k EQP requirements

For other passenger not needing a challenge, the M/S fare came GBP80 cheaper at GBP1353 , so the fares average out at GBP1393