The room setup being weird is the thing that strikes me about this. My desk (probably like many?) is against a wall, not a distance from it. That dead space between back of desk and wall with the projection on it isn’t going to work in many rooms, is it?

Desk against a wall could be handled very easily with a short throw projector but you would need to ceiling mount it upside down. The bottom edge of the projected image would trace back to the projector at a steep angle, your head would therefore not cast a shadow over the image.

Yeah, desk against the wall is a common setup (except in open-plan offices, where it's desk against a desk). However, when setting up my current home office, I opted for a different layout. It's similar to this:

https://www.ex-astris-scientia.org/gallery/other/defiant-rea...

but with an L-shaped standing desk from IKEA, with windows on the right side (behind the auxiliary desk), doors on the left, and remaining wall surface filled with random assortment of cabinets and wardrobes.

Having the desk in the middle doubled the usable wall surface area I could put closets and cabinets against. Also, I have something behind the screen for eyes to refocus on, reducing the strain.

(Also, I was totally going for the "Starfleet ready room" style; that's as close as I got. It's really hard to capture the feel without a proper carpet; unfortunately, I have dust mite allergy, so that's off the table.)

You will have to rearrange your room. It shouldn't be totally infeasible, though.