Well, it turns out it wasn't a proximity magnetic effect, it was plain physical interference.
Although it could only be heard in that split coil mode, and wasn't picked up by the other pickups, the four middle strings were just lightly buzzing on the GK pickup magnets, which were that bit too close. You have a very narrow window with the GK pickups. over 1mm from the string and the influence of the neighbouring strings is too strong for a clean signal. Too near and the strings will touch.
I expect when they touched or came very close to the magnet, they themselves briefly became magnetic pole pieces, much likes the slugs in a humbucker do. Which then upset the signal picked up by the split humbucker, causing the signal weakness.
Anyway, simply raising the strings a touch and lowering the GK pickup a bit cured the problem. In messing about with the GK mounting height, I did away with the small neck shim, and added some fixed height 'washers' made out of old plectrums, which I first drilled a hole in to fit the mounting screw, and then cut a piece out around the hole with scissors to fit under the ends of the GK pickup. (A couple of years ago I had ordered a mixed bag of Jim Dunlop pickups of different thicknesses and materials to see if I could change from my standard red sharkfins. I settled for the celluloid thins, leaving a whole load of unused plectrums, which make ideal thin spacers).
After this was done, the humbucker now sounded like a humbucker should, and with a lot more output.
It was then time to fit the Hipshot Tremsetter. A bit of measuring, a couple of holes drilled and the Tremsetter was installed.
The Tremsetter really keeps the bridge stable when you don't use the trem, and you can palm-mute quite happily without affecting the tuning. The two-point trem is floating, so the whole set-up really benefits from having it. If you use the trem a lot, I can really recommend them.
You can also see the graphite shielding paint in the cavity which neatly hides all the bits of repair work, plus it adds a bit of extra shielding. It's grounded by the ground lead running to the trem claw screw (which obviously also grounds the bridge and strings).
The observant among you may notice the small mark on the bottom left corner of the trem spring cavity. This is where the sixth of the trem spring cover pate screws wanted to go, but there wasn't any wood. The trem is an OEM 2-point Wilkinson unit with a steel block and steel saddles. The block has the now-standard Wilkinson offset string loading holes, designed so that the hole is in roughly the right place for the saddle position so that all the hole-to-saddle string lengths are about equal, and the hole isn't covered up by one end or other of saddle, but always comes up through the slot in the middle. However, this means that with a standard trem cover plate with either one big slot (as on this build) or six smaller 'racetrack' slots, the fore/aft sting hole distribution only allows one position to put the trem cover plate if you want to access all six holes through the slot. Which meant that one screw didn't work. So the cover plate now has a cosmetic screw head glued in that location.
So hopefully I'll take some photos tomorrow (Saturday). Steve/Arkieboy comes to collect it on Sunday. Provided he's happy with the guitar, I should then video him playing it in both guitar and synth mode.