Stabilize 12V to 12V

This looks like a good target for a SEPIC or Buck boost converter, which can allow an output voltage higher, lower or the same as the input voltage

Zener diode only limits the voltage. If the output is 11.6V, the zener diode will not do anything.

You didn't specify your current requirement, so its hard to give you a part as a starting point.

A boost switching regulator can increase the battery voltage high enough for a buck switching regulator to bring it back down to a regulated 12V. It won't have the efficiency of, say, just a large capacitor, but it will be regulated as closely as part specs allow.