| Maybe you're hitting the pup's magnets, or your is hard enough for the string to hit the fingerboard.
Another theory... look if it happens when you're playing an open note first, and then you press a note on the fret. If is that, the explanation can be that the click you're hearing is an electrical noise: as you play using a pick, you're not sitting your hand constantly on the strings, so if you play an open note and then you press a string, that's the noise of you reconnecting the "ground" of the circuit.
Why it only happens with that bass? Probably a problem regarding a proper shielding of your instrument. | 
