I don't have any way to test this without creating an account in your app, which I've really no interest in doing. Do you have a test login of some kind?
The "invite friends" link should get translated into this call to your server:
The "signed_request" bit triggers the WPBook plugin (lets us know we're inside FB), the is_invite bit tells WPBook to load the invite form, and the fb_force_mode bit tells FB this is an FBML page inside an app that is otherwise iFrame based.
I think the problem is that to draw and FBML page, Facebook itself is interacting with your blog - not the logged in user's browser, but Facebook's HTTP agent, which has no account on your site.
In short, I don't think the current invite page will work for blogs requiring a login.
That said, Facebook is deprecating the FBML method and now encourages use of the Requests Dialog - http://developers.facebook.com/docs/reference/dialogs/requests/ - so we may soon revisit the whole flow and that might work better for your use case.