4 LED monitors? That's half your bugdet right there. Also, for FPS gaming, always go for an odd number of monitors, otherwise aiming will be weird since the sights will be separated. CoD on max settings isn't too taxing so your budget should be fine (assuming you go for 1 monitor). Do you want to build it or buy pre-made?
Nice idea, except there is a glaring problem, what happens if you place more an two doors? I have seen a mod that involves teleportation golems, and the way it worked was, the bottom block of the golem worked as a "channel." it teleported you to the golem with the same bottom block.