If the regular bulb works fine as you said, it is not wired backwards
..
From your test, you know power is bottom, ground is body on both bulbs, and the AA courtesy light kit is wired the same way.
LED just is not getting power or ground from the socket, or, as was mentioned, insufficient voltage for LED, which I only bring up because you mentioned you are not using a battery so I assume you are using a battery charger for all this and that is what the alligator clips are? Just an idea there.