Well, there's a change in plans.
The initial reason for lowering the projector's box was because our house is designed to have a certain look and style. The outside is meant to look like something out of a fairy tale and the inside, for the most part, carries that theme through. So if there were a box hanging down in a notable position, it would look seriously out of place. A small box on the ceiling is easy to miss or to disguise, but one hanging down about 16" below the ceiling just doesn't fit.
She spent time on Pinterest and came up with an idea of making a small set of shelves hanging from that spot for knick-knacks, so we'd have a few things on display and it'd make that area look like it's an intentional part of the design.
I hate to quit or back out of a project, but I guess I'll have to save this kind of work (which I really want to learn about) for another project. I really can't argue with the simplicity of her idea and, considering how much work we're doing to finish things up, I have to admit I'm relieved that this is something I can just let go for the carpenter to do.
Thank you for all the ideas - while I won't be using them here, I'll keep it all in mind for anything similar I have to do along the way.