So Thomann agreed to give me a weirdly specific 99 € discount on the LTD KH-V, which puts me into the same price as their previous sale on the model which I just barely missed. Maybe they are running at pretty slim margins when getting rid of the stock for these, considering the model was like 1800 € originally.
I'll take it, because I already fixed at least some of the issues.
Jack plate could be bent back in shape and it seemed that the main issue was that the battery was not connecting properly. Interestingly it put out sound like that, just weaker output. I bent the jack plugs a bit and then it would work, and the scratchiness of the volume pot somehow disappeared too.
I think the tone pot still doesn't work, or is extremely subtle. I remember the one that I tried in a store worked normally.
Looking into the cavities, this does not have any of the quick connectors EMG normally has, just soldered. Pickups might have a plug connector. Couldn't spot any obvious fault with the wiring, but can't find a wiring diagram that is not using quick connectors. The pots are also PCB type instead of the normal lug type (little holes to put wires through).
Yesterday I was rocking the repaired KH-V through the Mark V 90, and I absolutely loved it. The narrow neck (42mm nut width) feels really good to play and it sounded glorious. Neck/middle position also sounds good for cleans.