After several deadends, it seems that there won't be available university positions in Beijing until September. There are a number of private school options, however those all require Sunday classes. I'm no Eric Liddell, but sabbath's not a day for playing ball.
So, for now, the plan is to stay in Cambodia (maybe Thailand or Vietnam) after certification. And I'll be looking for weekday teaching positions in Beijing opening with the next academic year.
Hopefully, this won't complicate obtaining a Chinese work visa. We'll see.