Can you ensure that you have the "Enable Automatic Resizing" option set to Enabled. That should try to 'widen' the module when it detects the send button has been clicked. It's our work-around for Facebook's implementation, and it's not perfect, but it works in most cases. If that doesn't work, can you post the URL to the invite module, and we'll take a look to see if there's another issue we can help you fix.
Thanks!