Erwin,
We've done quite a bit of testing on the front-end of your site. Admin access wouldn't really help us at all from what we've found, just so you know.
Your site or your application looks be affected by an uncommon bug from Facebook itself. The bug is that when the user logs in, the 'event' that JFBConnect listens for (auth.login) is not ever being called by Facebook. Because of this, the user is logged into Facebook, but not your site. When the page reloads, the auth.login event is triggered, which is why the user is logged in then (if you have "Automatically log Facebook users in" enabled). The original report for this issue is on the page below, and it describes your problem exactly:
developers.facebook.com/bugs/292597207451952
I'd highly recommend using the "Does this repro describe your issue?" link and add information about your site and application there.
Unfortunately, you can see this bug was originally reported in December of last year. I don't know what the timeline for the fix is, but I wouldn't hold my breath right now. Since your application isn't live yet, I'd honestly recommend trying to create a new one in the Facebook Developers Area and try to use that one instead. With JFBConnect's Autotune, it should be a pretty quick process to setup. There's no guarantee that that will fix the issue, but it's certainly worth a try. There's no details in the bug report as to why some apps are affected, unfortunately, so there's not much more we can go on.
If you do create a new app, any users that have authenticated your app will need to re-authenticate when they visit again. Since your app isn't really public yet, that shouldn't be a problem.
Please let us know how that goes, and good luck!
Alex