...Shielding could help, but the bass did not need it originally....
Your bass also didn't have this noise problem originally
Here's the thing -- if touching the strings kills the noise, then it's unlikely to be the pickups. Consider: the strings are between the pickups and any noise sources in the room, and they are properly shunted to ground, so it would be tough for room noise to get to the pickups without being "caught" by the strings.
I'm looking at a pic of ripper electronics -- assuming yours are the same ([Invalid or Expired Link Removed]) you could test that the foil backing of the pickguard has continuity with the ground lug of the output jack -- to insure that that foil is also shunting to ground properly.
All that said, there do appear to be unshielded wires that are a bit exposed in there, and there's also a component in there I don't recognize (Bill Lawrence designed these basses? I wonder if it's something similar to his q-filter, which is also a passive tone control that looks a bit similar) -- I wonder if that could be a culprit.
Personally, I'd decide if I wanted to keep the bass as close to stock as possible. If so, I'd be investigating every wire and path for continuity, and the ground wires, the current shielding, etc. If not, I'd reshield the whole cavity -- that's been good to me with all of my instruments, so I do it as a matter of course. here's a a pretty minimal thing you could try (a fair amount of labor, but zero changes to the wiring itself): take all the components off the pickguard. Line the entire underside of the pickguard with aluminum foil, making sure it has electrical continuity with the existing patch of foil (copper tape with conductive adhesive would be the best IMHO but I use tin foil on my cheaper guitars). Put all the components back on. What that does is give you a broader wall of protection to all the unshielded pickup wires.
Also, confirm whether the problem exists equally with all of the pickup settings -- neck, bridge, both, etc. If it's only when a specific pickup is engaged, well, maybe that IS the bad guy. They're both humbuckers, so they should each be quiet by themselves. If the noise is the same (or mostly the same) in all positions, then it's likely something in the electronics.
cheers!
ltt