You need an off-the-shelf (download and installable) server software that reacts in ways you want or to write such a server software yourself.
pagekite might have a setting to disable the “Temporarily Unavailable”. Otherwise ask the support.
Another option would be running the webserver on a non-standard, custom port such as 81 or so. Then even if not serving content (just a default html file) it could be abused for revealing if the workstation is reachable or not.