Have you checked the nuts & washers used to hold the controls to the pickguard? How about the neck screws/ferrules?
I have done all these things. I changed out all the controls because this is my 2nd Kiloton--I have a Fullerton as well. So to make this one different, I put a Seymour Duncan 2-band active EQ in it. I also swapped out the volume control for a B250k. So I know the nuts and washers holding the controls in place are good and tight. And the bridge saddles are all just a little higher than most of my other G&Ls, so I removed the neck to check for a shim. There was none, but the neck screws and ferrules were all nice and snug, and are once again.
Definitely WAY out of the ordinary, but I'm looking for a resonant vibration that is coupled to the body. Knowing how G&L does cavities (even Tributes), there might be an excessively long lead vibrating.
Since I swapped out the tone control for an active EQ and the volume control for a different one, I've spent some time in that cavity. I've seen my wiring, and it's not pretty, but it is effective. I did tie up the wires with small cable ties. I will try to very gently stuff some foam in that cavity because those Seymour Duncan preamps use very thin, brittle wires. Because of that, I have stopped using them altogether for my projects, though I know from experience they will handle an MFD pickup. I just happened to have one in my inventory, and it was the only concentric 2-band that I had. This G-string situation, however, was present when I first got the bass and before I installed the new electronics. Certainly worth a try, however.
Thanks, Ken!