Does the car have a clock ? working or not?

Also being an ex cop car it could be a left over energized relay that is not hooked to anything. Do you have any relays that are still in place ?

Also I would use that DVM hooked up in series (reading the draw) all the time when you are testing.

Alligator clips and test leads with clips are handy for jumpers and are cheap. the test leas also won't handle a lot of currrent so they can act as a fusible link as well AMHIK LOL