I'm not really sure why you need WOL. WOL stands for Wake-On-Lan, which keeps a small amount of power flowing to the network adapter chipset and some other essential peripherals. You can send a packet to that adapter's MAC address (there's no IP in this situation, mind you, since the computer is powered off), and that packet can turn on the rest of the computer.
I suppose that feature can be used to power up a PC, but then you need some other device that monitors your car ignition and sends that packet. Why not just buy an automotive power supply that has a startup/shutdown controller and connect it to your FitPC? I realize that it might not have external power button leads, but you could open the case and find which two leads get shorted when you press the power button. Also, a lot of FitPCs can automatically turn on when power is supplied, but that doesn't resolve a problem of monitoring ignition to turn the computer off when the car is off.
As for the battery drain, I'm no expert but I'm pretty sure car batteries aren't made to power low-power devices for extended periods of time - it will ruin them. Car batteries are made to give a lot of current very quickly to crank the engine during the ignition phase. Maybe I'm wrong about that part though, and someone else could chime in.