It's not a buzz, like when you touch the strings or let go, so grounding doesn't seem to be the problem.
The sound is similar to what I've heard described before -- "like tapping on a microphone to see if it works." Sounds microphonic to me. HOWEVER, the noise is only apparent when I tap on the chrome pickup cover I installed. When I tap on the pickup itself, I hear very little tapping noise, keeping in line with many of my other basses. (In fact, the bridge pickup on my EB-3 and the pickups on my Cort -- which I know have jimmied-up mounts -- make louder noises when tapped.) I also don't get noise when I tap on the pickguard next to the pickup, on the body, on the controls, etc. Just when I tap the pickup cover.
Any more ideas? FWIW, this particular pickup is quite hot, easily the loudest passive setup in my stable. And did I say I love the sound of it???

So I'm not inclined at this time to replace it.