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How difficult is changing the nut? I have similar strangeness on a PRS se - I’d used an old string to “file” the nut and put a little graphite lube in which made some improvements, but I still find the tuning and intonation quite unstable.
I was going to just drop the thing into a professional, but if nut swaps are relatively easy - actually, it’s more that if nut swaps are reversible, I.e. always retain the ability to back out and never cause permanent damage, then I’d give it a go.
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I had a guitar tech fit it for me. Looks pretty easy to remove and replace. Although you'd want the right tools if you need to sand the sides of the nut or file the nut slots.
skip to 4:30:
https://www.youtube.com/watch?v=qlHSrFx8iaM
Boring Tele setup update. I was mulling it over and had an idea that the neck could have been fitted at a wonky angle, therefore effectively making the length shorter on bass side and longer on the treble side, but in the end @ffm was right with it being a problem with the nut
Tusq nut now installed, setup perfectly and the strings sounds so much better.