One way is
scene.add(camera);
...
camera.add(bg);
Thus, your background plane will be a child of the camara and visually won’t change its position when the camera moves. You have to put that background plane far enough though, so the plane won’t hide obects in the scene.
Used this trick here for background gears: OMNIVISOR [mechanical eyeball]
Another way is to use scene.background = some_texture. The texture here can be RenderTarget.