Unfortunately, Facebook's invite implementation is.. lacking. The invite module creates an iFrame to Facebook which, once an iFrame is sized, it normally doesn't resize on the fly. When you click "Send invitations", the window grows, and therfore, needs to be resized. We have no control over the contents of the iFrame whatsoever, so we can't just move the popup to the left more. We do have some javascript (the "automatic resizing" parameter) to try and fix these issues, but it doesn't always work in every site configuration, and your site seems to be included.
However, it looks like in your case, there is a possibly easy fix. If you can modify the following file, it looks like it should fix the issue.
/module/mod_jfbcinvite/tmpl/default.php. At line 42, there's a long string. If you can remove the 'overflow:hidden' section, it looks like the module will resize properly for you. The final line should look like:
<div style="position: relative; top:0px; left:0px; width:<?php echo $closedWidth;?>px" id="jfbcinvite">
Try that, and let us know how it goes. Good luck!