Moving it to the interior of the car will help, and underseat installs are nice because they get the flow of air conditioning (since cooler air sinks).
I wonder how many of the problems you're having come from using an inverter. If you use a lower-power-draw motherboard, like something running an Atom processor, you need far less power, and produce far less heat in whatever case you choose.
We've progressively introduced lower-power-use boards in our systems, and they seem to get more and more reliable. I don't know if there's a direct (inverse) correlation between power usage and reliability, but that's what our experience indicates.