Plan B - Do it all direct with Emeco by email. Send Mrs Kamal the itinerary you need and she will make all the reservations (or be happy to leave it open-dated as you wish). In my experience you will need to email her also scans of your credit card, and it is probably worth calling her briefly once the first time round just to make sure everything is OK. She will be happier if you have a transoceanic reservation on AA (although of course there is no reason why you couldn't change that later, and it needn't cost you anything if it is a transatlantic that BA also flies).

If you want to do an overnight turnround at CAI then she will arrange for someone to meet the flight with the ticket and the credit card form for you to sign.

I am not aware that Emeco discounts from published fares.

And I am based in HKG, not Oz, but I have made arrangements with like this with Emeco three times now (after doing it once before with Mrs Sobhi at BA Cairo) and often been on the road when doing so.