I fully understand that this may not be a reasonable option.

Unless you have a huge room, I'd never build a home theater around a projector these days. I'd ditch the projector, even a medium quality LCD will provide better results than most home theater projectors. 86" LCDs are affordable and while you may sacrifice some size the viewing experience is going to be better.

If you stick with the projector don't notch the floor joist, lower the projector and screen if necessary.