You cannot save the numbers that the authenticator generates. Those numbers are randomly generated (every 15 seconds or so I believe) so storing them won't work. The only thing you can do is store the serial number on the authenticator if you lose it so you can get a hold of billing to have them remove it.

I am not sure how a single authenticator code works across multiple accounts for you unless you don't have a battle.net account yet. It never lets me log in to multiple accounts using the same code.