Hi Phil,
Having spent some time looking inside a Honer Jack recently (https://www.basschat.co.uk/topic/481089-honer-jack-electronic-repair/) I suspect the source of your noise problem might be the decidedly messy original wiring and the lack of any control cavity shielding - it really is a huge mess of over-long wires in there.
To answer your specific question about bridge grounding, in "my" Jack there is what appears to be an earth wire heading off to the bridge, so I don't think un-earthed strings are your problem (although having just checked the continuity with a multimeter, there's a about 300Ω resistance between the bridge and earth, and that's a bit of a surprise - not sure if it's supposed to be like that). Anybody?
And while I'm on, back to my recent Honer Jack post, if you were able to take a couple of pics inside your Jack to show me where the red wire from the battery goes, that would be very much appreciated.
Cheers
Phil