Seriously Bonernow, with your background you should hold out for the Boeing. If you only had Airbus time, you would have less ground to stand on - but in this case you have heavy Boeing time. I would not fly the A320 with your background - better to pass and get a job flying the 777 with EK or EY instead if you can get those jobs... I've heard that EK looks favorably on heavy Boeing time and QR is aware of that...
Plus, if you want to fly the 787, I hear you would have a greater likelihood of flying it if you were flying on the Boeing side of QR. That is what I have heard (could be right or wrong) - regardless, flying the 777 would be just as great if you had to sit on that aircraft. With your experience, you should get a slot on the Boeing side.
So, why would you want to fly the 787? Looks like the initial cadre of ANA pilots for the 787 really like the airplane. Check out this recent 787 video from Boeing:
Boeing Feature Story:'It's Fantastic'