How do you power a circuit that has a solenoid without heating the solenoid up?
the only use of the solenoid in the project is to "kick" the plunger
  when the circuit is cut off. So, essentially, I don't need it for a
  continuous use; just for the moment when the circuit is switched off.


If no other power source is available to operate the solenoid when the circuit is switched off, you will need to store enough energy to 'kick' it when power is removed. You can do this with a capacitor which is charged up when the circuit is connected, and discharged into the solenoid when the circuit is disconnected. Here's one way to do it:-







simulate this circuit &ndash; Schematic created using CircuitLab

When S1 is closed (ie. circuit is switched on) relay RLY1 operates and connects capacitor C1 to 12V via current-limiting resistor R1 and diode D1. C1 then takes about 2 seconds to charge up to +12V. When SW1 opens (ie. circuit is switched  off) RLY1 releases and connect C1 to the solenoid. C1 then discharges its energy into the solenoid, causing it to 'kick' until the capacitor has discharged. 

D1 prevents the voltage on the capacitor from keeping the relay operated when power is off.

The value of C1 required depends on the solenoid's resistance and how long it needs to be operated for. With the values shown it should operate for ~150ms (assuming that once operated it holds in until voltage drops to 3V).