If it kicks in momentarily when the starter engages, it sounds like a bad connection somewhere that "makes and breaks" as the engine is cranking. I'd check every power connection you can find under the hood, make sure the battery cables (positive and ground) ends are solid, check the ground cable connection to the block, etc. I don't have an electrical circuit diagram in front of me and I don't know how your AirDog is powered. But I'd trace the wiring back from the pump to wherever it's connected and begin my diagnosis there. It's probably operated off of a relay. Check every ground you can find as well.