After advise from Ashley I contacted the Awesomium team regarding the performance bug in fullscreen mode.
This is their answer:
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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
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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.
Cheers!
Btw. where is my badge...