I can almost envision how to use a web server to handle the issue.
Using http calls, you can invoke the Facebook login page which takes an input of another web page to go to (passing Facebook information into it) after the login is complete. Then, assuming the actual app window title is always the same, you might be able to use some Javascript to bring the app window to the front. I'd have to try a test to see if that is possible (the javascript part). If I can make that successful, then it should be possible to use http calls to Facebook.
Within the app itself, it should have a function to handle the return from background call. If it is possible to bring the app window to the front, it will then get one of those AppGameKit call backs. The function called for when GetResumed() returns 1 (something that should be checked near the top of your main loop) can check a variable set when you send off the http call to decide if it has just come back from a successful Facebook login.
Like I said, I can visualize the entire sequence. If I can find the time, I will try a couple of simple tests and post the results, if it works.
Cheers,
Ancient Lady
AGK Community Tester and AppGameKit Master