Not sure if it is worth it for your use case, but I was successful in creating a virtual display - second screen - and putting the window there. That did work. I was trying to get kb input and mouse clicks to work, but then realized that that was even more useless than spinning a window to begin with (lol) so backed out all that effort for this much simpler implementation.

EDIT: private APIs — CGVirtualDisplay / CGVirtualDisplayDescriptor / │ CGVirtualDisplayMode (the same ones BetterDisplay uses)