Well you'd need to hold down the button on the dash for several seconds for the PSU to fully power up, and you'd need to use diodes to seperate the car mains from the +12V rail on the PSU (on both + and - since the PSU ground is seperate from the car ground)
But yes, I'm no electronics expert but that sounds like it would work for your situation.
Basically you use the power door lock relay to trigger a ATX power switch pulse in the hopes of turning the computer/PSU off.
You use the +12V from your PSU to keep the relay closed for the power to the inverter, and you have a pushbutton in the front that will momentarily give that relay for the inverter +12V from the car to close it until the PSU can power on, at which point the PSU maintains the relay closed.
I think if you use 4 diodes to isolate the car and PSU power from the inverter relay you should be OK. The only problem that I can see is the computer might not always power itself off when the power door locks press the power button, there are many cases in which it might put up a user prompt or something preventing shutdown. In this case you'll drain your second battery, but as long as it is seperate from your main battery then I guess it's not that big a deal.
You could also load a program on your computer to force a shutdown once the power button is pressed, so no matter what prompts come up it will always shutdown. This should limit the cases when it might drain your battery.