Simple piezo buzzer circuit--could I power it using a large battery?
You have basically two kind of piezo buzzers :


Basic ("external drive") piezo buzzers are a simple piezo disc (a plate of piezo material tied to a metal plate), usually mounted in an acoustic resonator. As  Oleksandr R. said, it behaves mostly as a capacitor so if you power it with DC, you won't get any current through it and therefore no power (and no sound output). Those work by applying a (fairly large) AC voltage, and they work best if the frequency of the voltages matches the resonance frequency of the piezo disc, typically a few kHz.
"Self drive" piezo buzzers contain the necessary oscillating circuit in addition to the piezo disc. They take DC voltage input, and do the AC conversion at the optimal frequency before feeding that to the piezo disc. The "12V" you mention seems to indicate that it is this type you have.


In both cases, if it's a "12V" piezo, I don't see any reason why you souldn't plug it to a 12V voltage source. If it's the first type you'll hear a tiny "click" (the AC transient as you plug the DC) then nothing ; if it's the self driven type you'll hear it sound loud and clear.