If it works with detecting the lights - and turning off everything (but the phantom) results in a meter reading of 0 amps, then you can be certain it will detect a few amps current. At that stage, I stop using the clamp on and pull the battery cable - then use the meter leads on a 10A scale - with the leads connected between the battery and the removed cable.
That will easily detect low current draws. Just be sure you don't inadvertently turn on something or try to start the vehicle, which will blow the meter due to current overload.
(I don't recall if the Sears meter has direct current detection with the leads - but many meters do).