From what is described, does sound like render selected (on its own at least) is not the solution. Depends on the purpose, to save render time, or to allow editing the object independently in post? If the second, then using some object mask render element should let the object be isolated in post, with no need to change the render.
If it's about render time, not sure - region render as noted could work, depends on whether the object will be moving around its location within the frame as not sure render regions can be animated. If not, then another possible workaround is using a plane with 0 roughness, 1.0 IOR, full refraction, and use that as the sole render selected object. Since "what it sees in refraction" is fully maintained as part of its render selected, this plane then becomes a "window" through which only that part of the scene is rendered. Most likely this would need to be right up close to the camera (else the depth of the fog may be different). Not the easiest of set ups, but you can then at least animate that plane to follow the camera, and even to move around in the frame to always keep the desired selected object in view.
EDIT - crude test scene attached in case the description of that last set up was somewhat confusing in text form :)
