In the meantime, I disassembled the bass. The guy who "improved" on the bass glued a piece of thick aluminium foil over the electronics cavity. When opening it up, the whole circuit fell into pieces. After that, I didn't get to looking at it for a week.

Yesterday, I spent an afternoon with a friend and tried to realize a circuit (with the aid of the schematics on the Seymour Duncan website). No matter what we tried, we still kept getting a hum (even when directly connecting the pickups to the socket, without any potentiometers). It sounds like there's a grounding problem.

So now I'll have to remove the complete pickguard and check the pickups. *sigh*