First I would like to personally thank all 15 people who viewed this post.
In case someone finds this while searching the archives, I got some answers over at Apple's discussion boards. As I feared, you can passively share an Internet connection over Ethernet, but you can't actively control it; you need an AirPort card for that.
There is a
third-party utility that would do this, but I'm not that adventurous.
Another work-around would be to set the base station to automatically connect to the Internet, and make sure it's set to automatically disconnect after x minutes of inactivity.
But as for me, looks like I'm in the market for another AirPort card.