I'm not sure I can help you, but I'm also not sure I understand your request entirely. You don't need help with the projection end of it, do you? (idk anything about that.) I mean, all you need is a chess program that displays on a PC monitor, right?
Also, I'm not sure what you mean about being able to move the pieces around freely. Will there always be both kings on the board, and will you always be making legal game moves, or do you also want to make illegal moves? (If it's the latter case, I only know of two programs that will allow that.)


Need some help!! I am helping teach chess at our elementary school. I am curious if anyone has an idea of a software program that I can easily setup diagrams and be able to move pieces around freely. I will be showing this on a projector. Please, any help would be awesome!!!!