Thank You, I don't think you realize just how many people you have helped, mainly because there are so many of us who use twentytwelve child themes, and for me, the menu on every single one of them just broke with that update, not just the ones with a second menu. I have been trying to figure out what changed, but was looking in the js and the style changes that I had made, rather than the header.
Based on what you just spelled out, all it will take is to change that h3 to button in the header, and all are fixed for those not requiring changes in the js itself.
I would like to make a suggestion though, and maybe you are already doing it, rather than making changes to the parent theme js, copy it to your child theme to make your changes, then dequeue twentytwelve nav.js and enqueue childtheme nav.js. in your childthemes functions.php.
This will avoid overwriting of the changes in future updates.
Cheers and many thanks!