First, you probably have no need for a backup battery. The vehicle probably already has a battery, to power the starter motor, and it has a LOT higher capacity than what you need. So use it.
Second, if you restrict the domain of discourse to "cars" instead of vehicles, a very cursory review of basic automotive electrical systems will show you that there are basically three electrical networks: a momentary relay-switched very high current system (starter motor current), a switched low-current system that is only active when the ignition circuit is switched on, and an unswitched low-current system. The switched low-current system powers the ignition system and the vehicle electronics. The unswitched system powers things like courtesy lights.
Finally, you're over-thinking this. Go back to Electricity 100 (not even 101), and do it the way they did it in the Dark Ages, before silicon was discovered.
A capacitor and a relay are all you need. The relay contacts connect the unswitched low-current leg to your computer. The capacitor goes directly across the relay coil. (You probably want a reverse diode across the coil as well, to snub the reverse kick from the coil.)
When the system is sitting there, dead, the capacitor is discharged, through the relay coil.
When you turn on the switched leg (ignition), the relay coil is energized and the relay pulls in, applying unswitched leg voltage to your computer. The capacitor charges at the same time. As long as the ignition is on, the capacitor is kept fully charged.
When you turn off the ignition, the capacitor starts discharging, through the relay coil. At some point along the discharge curve, the voltage on the capacitor goes below the holding voltage on the relay, and the relay drops out, removing power from your computer.
Use the known resistance of the relay coil, from when you chose the relay, to size the capacitor to give the delay time you want.