I'll write more later, but the happy news is that my F-4 visa application went without a hitch. I spent a lot of money, but in two* weeks, I'll have my new alien-registration card and an F-4 visa that'll be good for two years.
The point of this visa is that it's not attached to a sponsoring employer: I essentially become a free agent, able to roam Korea at will, working wherever I want, enjoying most of the perks of a South Korean citizen thanks to my Korean heritage. If I suddenly want to act in a TV commercial, I can do that while working elsewhere. Private tutoring? Also no longer a problem. No need to look over my shoulder for the authorities.
After a raft of bad news, this is very good news, indeed. More later.
*Corrected. I'd originally written that it would be three weeks.