I'd still go the way of connecting to VGA or DVI. A craphics card with two VGA outputs (or one DVI and one VGA) could be used to display the VI on the computer on one screen, and the captured image in full-screen on the second.
I have no experience with telling LabVIEW to use one screen or the other though, Maybe someone else can help here..... If the projector can accept files (*.jpg, *.bmp?) then perhaps giving the name and model of the projector might help..... Greets Shane.