Oliver,
Very sorry for the delay in this. To make the Invite module always appear is pretty easy. However, as stated, it will always appear, whether the user is logged into Facebook or not. There's no way for JFBConnect to detect if the user is logged into Facebook at all, we can only detect when a user who has approved your application is visiting your site. Therefore, the code change below will show a blank Invite module (no friends) when a user isn't logged into Facebook. Please test, and if you have suggestions on how to improve what you get, definitely let us know.
In the /modules/mod_jfbcinvite/mod_jfbcinvite.php file, at line 21, simply set $fbUserId equal to true, instead of what's currently there:
Let us know how this goes, and good luck!