Playwrights or theatre designers -- Can this even be done? Inexpensively?

Projecting images (whether moving or still) is not difficult at all - we do it regularly. I think it's great that you are considering the practical aspects of your play - I've worked with writers who don't (did a new play last year which was virtually a film script, calling for some very tricky effects, and a play this year which had been adapted from a TV script, again calling for some tricksy stuff) and writers who do - and I know which I prefer! What you're proposing doesn't sound too complex and I'd be surprised if people had issues with it!