I guess the image that came to mind for me with climbing was the one woman I passed on the road a few months back going up a moderate (but short) hill in her aero bars. Ok, that's fine, it wasn't a significant climb. EXCEPT she was also standing out of the saddle. That just boggles my mind.

Anyway, for a new rider, I wouldn't suggest doing anything over a false flat in them really, especially if she's afraid of wobbles, because the more power you're putting into the cranks, the more likely your upper body will do some swaying, and then you don't have the stability of a wide grip. Plus, you can't breathe as well as when you're more upright.