Not so

On my last xONEx I had three stopovers in PER, perfectly OK
(...HKG-PER-BME-PER-DRW-PER-HKG...)

In the continent of origin (in your case Europe/Middle East) you are allowed only two stopovers. You have the allowed two (both in London as it happens) so that's fine too.

As Moomba said - there is a bug in the tool that incorrectly prices the itinerary when you start in one Middle Eastern country and end in a different one.

With regard to getting your itinerary ticketed - you could book it with the AA round the world desk and then contact AA's sales agent in Cairo to pay for it. To do this AA insists on at least one of the "over water" (i.e. inter-continental) segments being an AA flight. On your itinerary that could be either HKG-JFK or LAX-SCL flight (CX or LA metal, but book the AA codeshare). If this option is of interest, let us know and we can provide advice on how to go about it.

Enjoy

ps You must of course consider the frequent flyer mileage implications of your trip- a whole new world of possibilities!