For smaller views (like on sidebars), you need to enable the "Enable Automatic Resizing" option within the Invite module. The description states:
Enable this if the module area to be displayed in is less than 582px. This is to overcome a bug in Facebook's implementation where the 'Confirmation' dialog box can get chopped off.
I'm assuming this will fix your issue. Facebook's implementation of the Invitation options is pretty horrendous and we had to invent quite a few workarounds. I'm sorry if you think your not getting your money's worth, but there was a large time investment in making thinks as simple as a radio button to overcome their issues. Please read the descriptions. If you have that enabled and it's not working, definitely let us know and we'll work through it with you.
Invited - alignment
