Oh definitely. There are pros and cons to invite-only, just like with anything, but if you don't like the idea of doing it then you probably shouldn't do it until you're down to your very last resort. I know you can't install scripts on IJ, but here's a suggestion that just sprung to mind (lolsprung): Host your application offsite and set it up to log IP addresses.
I know it sounds like a pain, but it would most likely eliminate your issue. If that sounds like something you wanna do, I can help you set it and the scripts up, and style it to match your game pro bono since it's such a bullshit situation. :3