I know you're talking about 12' from screen to projector, but is it possible that one of those ridiculously short throw projectors would do what you want? I don't remember any brands or model numbers, but they use a different approach to the throw and maybe this could help.
A good answer is easier with a clear question giving the make and model of everything. "The biggest problem in communication is the illusion that it has taken place." -- G. “Bernie” Shaw
