Here’s an outtake from the article:
Other awards received by Twilighters for non-”Twilight” performances fall primarily in the niche, festival, or independent realm. Robert Pattinson’s sole non-”Twilight” win was awarded by the Strasbourg International Film Festival—he was honored with a “Best Actor” designation for his performance in “How to Be”.
ETA: He was actually awarded Best Actor back in 20o8. Thanks RobPattOnline for the info.