After advise from Ashley I contacted the Awesomium team regarding the performance bug in fullscreen mode.
This is their answer:
We realize GPU accelerated rendering is a big issue for a lot of our users and we're now making it a priority in our 1.7 branch.
Best regards, Adam
So don't hesitate about contacting them about this issue, because they need to know how important it is to many of us to make it a priority in the long "to do" list.
Seems to work.
Btw. where is my badge...