Is it possible to just use a large bank of capacitors to keep the LEDs on? When the car is on, the capacitors will slowly charge up (and the LEDs brightness will slowly increase). When the car turns off, the capacitors (not the car battery) power the LEDs until they become empty.
Your LEDs are probably much larger than what I am imagining, so a capacitor bank for 5 minutes of power is probably not practical.
You could use a relay or transistor with timing circuitry like a 555 timer. I do not know anything about cars, but I suspect you can tap into the power wire that is controlled by turning the key partway (to activate the car's electrical system).