Ceiling – Connecting two joists with a wooden piece to hang a projector screen

Just because you can't see the joists doesn't mean you cannot use your beam approach for exposed rafters. The structure is essentially the same, except for a thin layer of sheetrock between the two, which has little impact on the forces involved. The best thing to do is distribute any concentrated load over multiple connections. Use a flat 2x4, or better yet 2x6 that's long enough to span 4 joists, about 4 to 6 feet. Place the chair connection at the center using a large through eye bolt, using an over sized washer to spread the load into the wood. The nut and bolt end will end up punching into the ceiling, but that's OK. Holes are easy to patch if the beam is ever removed, otherwise, it doesn't show anyway.

Attach the 2x beam to each joist with two 1/4" dia. lag screws. Use a screw long enough to penetrate 2 inches into the joist material, probably 4 inches. Drill a 5/32 inch pilot hole before placing each lag to prevent splitting the joist. The resulting 8 screws is more than adequate to support any adult and will not significantly weaken the joists.

Most screens have both screw-mounts and hang-mounts (loops) built in - the loops often fold if not in use. If your screen does not have the loops then you could add your stick/board to it to get some. Or it may have largish-holes that a chain screw-joint link could be placed through on the end-caps.

Use the hang-mounts and chains and screw-joint links as seen on trailer safety chains (or carabiners if you'll be moving it frequently, I guess, and it's not a public-ish location.) It's actually more stable if the hooks on the wall are wider than the hooks on the screen, so the chains angle in. Use chain, not wire. I speak from experience on that point - also use chain and links a good deal stronger than the weight of the screen would seem to require.