Wouldn't it be possible to divert power FROM a car battery, and not just charge it with one of those things? Just curious.

Probably not. There's probably a current direction circuit built into the charger. Go for either an inverter or cigarette lighter socket with a couple of clip connectors (like small jumper cable ends). I used to have one of those where the inverter could be removed from the main unit & had connector wires for just that set-up.