Okay, on a whim, I removed the function so that no Javascript replacement occurred, just to see if the default Javascript would do something different (never tried that, I just went with Zeak's tutorial "out of the box").
No change. Same problem. So know I don't know if this is because of the Javascript in either the default or modified Javascript, or if there's something in the way TwentyTwelve is set up. I'd be doubting it's TwentyTwelve because I'd think that adding a second menu is a fairly common thing and if double mobile menu problems were occurring, this would be talked about more often.