Olie; I doubt it's the pickup, but if you want to investigate further, you can pull the pickups out and switch them around. If the noise moves with the pickup, it's the pickup; if the noise stays with the same filter, then it wasn't the pickup.
Another thing to try is a different 1/4 cable.
On the Lenny recording, that's the Maple S1 tuned down a half step (I don't think I had the Walnut bass when we made the recording). I don't remember the tone settings on the bass. The preamp is an F-1X; the reverb a TC Electronic M-One; and then an SF-2 and an Ashly SC-50 compressor/limiter. A Bag End D10X-D and an Acme Low B-2 were both run full range; the Acme was on the floor (carpet), the Bag End on top. A single mic was place a few feet away; I don't remember what kind of mic it was. The SF-2 was setup in the mono parallel mode with one side being low-pass and the other high-pass. The cut-off frequencies were probably somewhere in the neighborhood of 100 and 600Hz respectively; and there was probably little if any Q. The F-1X was probably 5-5-5 or very close to it, with no deep or bright boost. On the bass, I usually have the neck filter closed with the Q off, and the bridge filter open with the Q on high; but on that song I doubt that was the setting, but maybe something close to it.