Thanks for sending me the link to the popup you see. I tested in that view and there are "Invite" links next to each user. That's not something I've seen before, but it may be a new test from Facebook. When I click the "Invite" button, it did send an invitation to that user.
Previously, there were checkboxes by each user and you could select a lot and then click "Send invitations". Facebook has been trying to cut down on spam, and I can see this as a way they are doing that as it prevents just checking all the boxes and hitting "Send". The system I see on your site makes sending each invitation a little more intentional.
With that said, you only send me the invitation window link. If you can send me the page from your site where I can see the link to that window (your JFBCRequest instance), that may help see other issues or problems if you're running into them.
If not, everything actually seems right to me.. just that Facebook has changed the invitation feature a little bit.
Thanks,
Alex