I would just feed it with the "accessory" output from the ignition key switch. That circuit usually is not powered while the key is in the start position, so anything powered by the delay circuit will not have power during cranking, but will get power for 3sec after the key is moved from the start to the run position.
In modern cars, lots of things (radio, AC, blower, some lights) are automatically turned off during cranking so that the starter gets the full available output of the battery...
Note that my circuit resets the timing network when input power goes away...