There is no difference between not found and unauthorized error pages/message, so they are essentially the same (that's how application are generally designed so that you cannot leak the fact a resource exists).
Therefore you could rename the "not found" message to mention that "the content does not exist or might require login" or similar.
Actually distinguishing between not found and unauthorized would require some additional logic with additional database requests.
There were some previous discussions about adding a way to login from the 404 page, this is currently not possible with a bare Flarum installation. The closest solution would be to use the HTML Error Pages extension together with the Direct Links extension so that a link to the login page can be pasted into the error page.