Yeah, races that don't accept foreigners are the bane of my existence. I've had Korean coworkers take care of that for me in the past, but it's annoying. I won't bother with supposedly international races that don't accept foreigners.
Also, it's ...
You can ask a friend or Korean co-worker to do it for you. Failing that, just try muddling around the site yourself.
The two groups I'm aware of are the Seoul Flyers and Smile Run.