The Relationship of Power and Heat to Length of a Folded Resistor
Yes, you will need to double the voltage to get the same power output if you double the length of the wire.

On the other hand, you could connect two 3-inch heaters in parallel and operate them from your 12 V source. The battery would be required to provide twice the current in this case...if the voltage drops at the higher current level then you won't get quite twice as much power.