Hmmmm... (No pun intended!) I have a very similar but even more complicated problem.

I have a 5 string fretless bass. I bought it off fleabay a couple of years ago, at a bargain price, as know the luthier who made it back in the 80s and who reckons it's, more or less, unique.

The pickup it came with , to cut a long story short, was knackered and my luthier friend told me he'd stopped using that brand as they always failed. He fitted an LR Baggs but when I got it home - you've guessed it - Hummmm. I tried all the usual things - check the earth, take it off, put it back but it made no difference. Back to his workshop - no hummm.

By bass amp (small poxy cheapo - I'm really no bass player) doesn't hum with my other (straight electric) bass plugged in and the fretless doesn't hum when plugged into a headphone amp. Touch the plug socket and the hum goes away. The hum gets worse when I go near to any form of digital equipment so I'm convinced its some form of peculiar induction between the electrics and myself.

I too would appreciate any ideas.