FWIW, I had a lot of trouble booking flights on Egyptair's website - it rejected my card (twice, two different bookings). When I called Egyptair in the US, they said that I would have to fax in a signature, accompanied by a copy of my passport and that they would email me a form for this, which they never bothered to do. I wasn't about to send them a copy of my passport, so I ended up booking the flights through Expedia (which allowed me to use my Amex - Egyptair's website only takes MC and Visa). I didn't realize I was going to get charged twice as much as if I elected "Egyptian resident" - but at least I managed to secure tickets.

Another option you could consider is just waiting and buying the tickets for the internal flight once you get to Egypt - even doing it on arrival. I've done this before for internal flights in Spain and saved about 50% of the ticket cost compared to booking online from North America before departure.