I used to install custom home theater stuff (some very custom).
What is the distance from the chairs to the wall (viewing distance)? Even HD will lose its "definition" if you sit too close to a very large HD image.
Some basic thoughts about projection:
As everyone has said, it's all about the lighting (and many other good points above too). Doing a projector, and doing it poorly, will be very unsatisfying. Very unsatisfying...
Doing it well, is awesome.
Does the room have any windows (or outdoor light coming in from other rooms)? And if it does, do you plan on watching anything during daylight hours (you really should have 100% blackout capability for windows, if you want to do daytime viewing)? For a projector setup, you really need control of ambient lighting. Window treatments may be a "hidden" cost of a projector.
And it doesn't seem like you haven't factored in the cost of the screen (and frame, most likely) to go with a projector (without a proper screen to project onto, don't even bother, it just won't look good). $800 dollars is a very limited budget. You'll need to overcome any ambient light emitted from your pins, if you'd like to have them turned on at the same time. So a darker screen would make sense (to darken the dark colors and shadows) + an even brighter projector.
One advantage of a projector set-up over a glossy-smooth TV surface some is you won't get distracting reflections (blinking pinball lights?). So there is that too. Where are the pins? Some TVs have a matte surface to help with this, but have their own problems.