Well, we get 3D scenes/projections on a 2D screen by being in a 3D world/context and looking at a 2D screen. With that I mean to say that I think (not sure) a 2D image of a 3D projection can only feel 3D if you're in a 3D world.

This also means such a thing should be possible for living in a 2D space and then projecting a 2D world on a 1D screen.

If you're going to play with 3D projections of 4D objects, then in order for it to really feel 4D we need to live in a 4D world. In that sense, this is quite similar to living in a 2D world while seeing a 3D projection. That's way more trippy.

To recap:

Feel 3D: 3D context -> 3D projection -> 2D screen

Feel 2D: 2D context -> 2D projection -> 1D screen

Feel 4D: 4D context -> 4D projection -> 3D screen

What we can do for 4D: 3D context -> 4D projection -> 3D screen

Compare issue with 3D: 2D context -> 3D projection -> 2D screen

I hope this makes sense (and I hope I have it right).

Man, this is tough to write about. I'm not sure if I fully understood all the nuances of your comment.