You could use set up power management on the mobo to reboot computer after power loss. This way you could run a wire off the ignition to a relay to send power from battery to invertor when ignition is on. When you shut ignition off, power would be cut and computer would go off. Of course, you want to shut down, hibernate, standby, whatever, not just cut the power, or you can damage your setup. However, when you start car next time, power would be sent to relay, and motherboard will recognize power and should restart your setup. You may have to play with the power management setup. The way I have mine, I can shut car off and still have power for pc. Pushing an extra button wasn't a big deal for me.
Another thing you may want to think about is that if this is wired with ignition switch, anytime you turn ignition on, computer will start. This could be annoying if you had to move cars around in the driveway, drive two doors over to pick up the neighbor, or any other times when you are turning car on for only a very short period of time.
If you don't shut it off right, you will eventually do damage.