If you ran a projector, it would have nothing to do with the BeoVision 8. I don't know what your history with B&O is, but consider it more like the MX than an Avant. Bang & Olufsen tvs seem to fall neatly into either televisions or home cinema systems, the latter being able to control a second monitor (like a projector), drop a screen, dim lights, and do all other sorts of magic tricks.
The BeoVision 8 is just a TV. An absolutely fantastic one I must say, but not a Cinema Master. To do this the proper way you will need a BeoVision 7, 9, or just a BeoSystem 3 with a BeoVision 4. Any of these will drive a projector and take control of the Apple TV and other sources.
If the other TVs are outside of your price range for now, don't shy away from the BeoVision 8. It, like all other Bang & Olufsen TVs, will at some point get bricks thrown at it on these forums because it doesn't make unicorns appear and cost the same as a Sony, but it's a gorgeous TV with the best picture available on a 32" screen, great speakers, and can still control your Apple TV and STB. You can even connect the BeoCenter 2 with Scart for full integration there as well.
Good luck!
There is scarcely anything in this world that some man cannot make a little worse and sell a little more cheaply. The person who buys on price alone is this man's lawful prey. - John Ruskin