Huh. Have you posted in the Luthiers Corner, or in the Pickups Forum about this? Might find someone with that same experience.
I've had pickups go bad before, open coil, shorted coil, but not one that behaved like yours.
I'd be tempted to pull the pickup out, have a look for anything askew underneath. Just for curiosity's sake.
I'll probably post. I don't care any more as far as this bass is concerned but I'm curious more than anything.