Okay, discovered a clue. The addtional menu has a different class in order to style it differently on my desktop setup from the main menu. So it needed to have all the CSS from the parent theme's mobile section with it's own class assigned to it. I copied all the mobile navigation into the child theme, and duplicated so there's a set for "main-navigation" and "upper-navigation" classes. (And made sure I added the .menu-toggle class set to "display:none" in the desktop areas as well).
And now I realize I was going to need to do this eventually so that I can style both the menus in mobile so they don't have the default TwentyTwelve look).
So both buttons are now closed. But they won't open. The investigation continues.