We're going to have to investigate this a little further as I don't have an easy solution to this offhand. One thing you can try, if you'd like is the following:
In the facebook.php file mentioned above, there are 4 lines that use the JRoute function. These really aren't required for JFBConnect, and are used to just make the 'pretty' URLs for your site, if you have SEF URLs enabled.
If you edit that file, and look for the 4 cases of JRoute, you can try to remove them. For example, one of the lines looks like:
$logoutLink = JRoute::_('index.php?option=com_jfbconnect&task=logout&return=' . $logoutUrl, false);Change this to:
$logoutLink = 'index.php?option=com_jfbconnect&task=logout&return=' . $logoutUrl;
Do this for all 4 cases, and see what happens. Most likely, a different Fatal Error will crop up, and it may not be as easily solved, but I just don't know enough about the synced mode to give a better answer right now.
If that doesn't work, we'll try to investigate a little further later in the week (or this weekend) to see if there's a simple solution.
Good luck!