Jan,
I don't know if JFBConnect is going to be able to do what you want. Facebook separates out Fan Pages and Apps in a number of ways. I believe, if you have your application configuration (in Facebook) setup correctly to point to your Site's URL (PS) then you can use embed that page anywhere, and when the user logs in, it will still be to that page. I can't guarantee that though, as we don't do testing with JFBConnect on Fanpages, and focus more on the 'on-site' implementations of Facebook Connect.
We do have a 30-day money back guarantee, and you seem to have some know-how on how things work. We'd gladly help you how we can (as we'd like to see some good working implemenations of FanPage/Facebook Connect integration), and if it doesn't work, you can get your money back.
Hope this helps a little, and sorry for the vagueness of the answer.
Thanks,
Alex