I'm sorry, that is what I meant. So few people aware of them that any time I read projector I just assume they are talking about LED or LCD projectors. My bad. However if that's what you have, why not build a box to protect it and either set it on the floor or mount it to the wall really low, or within a console that will protect it? Iis it just that you don't want to give up floor space?

Don't touch anything structural, especially doing it yourself. You can have someone come out and figure out a way to resupport around that joist, but at that point you'll be spending enough money where you may as well just buy a laser TV. Those are much higher quality than a projector anyhow.

In fact this would be a great way to justify buying one to your SO.