Phantom draw will be very low, probably <50mA range or lower. Then after the car sits "idle" for awhile (no activity through the computer for 10- 30 min) it will be even go lower than that. I'm not sure you will be able to see such small ranges on that meter.
Be sure to zero the meter with the jaws closed, and not around a wire. Then try the lowest range 0-20A to see if it can detect that small current. Clamp on meters don't do well when there isn't much magnetic field built up around a conductor unless they have an external sample loop.
Yes, try the experiment suggested by Drycamper11 to see if the meter will indicate a higher current. Headlights should be fine at the 0-20A scale.