Guy,
Very sorry for the delay. We're really not avoiding this. We have a plan for something to test that may fix this how you would like it to work. We've done very little testing with this, but would love your feedback. It's an extremely simple change to make and hopefully will fix up all these issues.
If interested, please edit the /plugins/system/jfbcsystem/jfbcsystem.php file. Around line 199, you'll see:
$subs = "\nFB.Event.subscribe('auth.login', function(response) {jfbc.login.on_login();});";
to the below (changing auth.login to auth.authResponseChange):
$subs = "\nFB.Event.subscribe('auth.authResponseChange', function(response) {jfbc.login.on_login();});";
We came up with this a few days ago, but since it concerns the login functionality (an obviously core/important feature) it's something we really would want to test more before recommending it. However, since you can obviously recreate the issue well and know exactly what you're looking for, your testing and feedback would help us greatly in this case.
Please let us know, and again, sorry for the delays in getting back to this!
Thanks,
Alex