Defining hook_menu() paths for specific base URLs
If you want to use base url, you should first declare it as a global variable and then use it as $base_url. You should also note that $base_url will include the url scheme (i.e) http or https.

If you have a development or staging environment setup, your base url will probably differ between dev, stg and prod environments. In most of the cases, the production environment will be SSL enabled. So you may have to consider this as well while writing code.

In my opinion, as you have mentioned, you may use a different module and enable it only for the specific site. Or if you deliberately want it to be in a common module, consider using access arguments and define hook_permissions.