It turns out that I was right. IE doesn't fire onChange when you click on radio buttons, but waits until you take focus elsewhere. After wresting with uc_shipping.js, I was surprised to learn that I could just replace the change() with click(). I was concerned that it wouldn't fire the event if someone used the keyboard to select the shipping option. My concerns were unfounded, and onClick works even better than onChange for dealing with radio buttons.

Revision 1075 has the fix.