Someone in the TEA thread mentioned it might be thermal drift, so I decided to see if that is so. TLDR; it isn't.
I took two sets of measurements at a tA of 19°C and 26°C. I measured at 1uA, 2uA, 4uA, 10uA, 100uA, and 199uA.
Aside from the readings drifting up by the best part of 1uA and getting a bit noisier, the offset remains the same.
Then I discovered something which might start to shed a bit of light on the matter, and I'm not ashamed to say, entirely by accident.
I was taking the leads out and switching the meter and calibrator off, and purely by chance I left the F87 on until after I removed the leads... and the offset almost disappeared!
A quick fiddle about reveals that without the banana plug (once I saw there was an effect I used a plug on its own rather than a lead) the meter reads 0.2uA in 600uA range, 0.07uA with HR, and 0uA in 6000uA range, -0.1uA with HR.
With the plug in, the readings are 0.0, -0.14, 0, -0.3 respectively.
So it looks a bit like the lead detection system is causing at least a part of the issue, maybe they thought to compensate for it in normal res mode but forgot to do so for HR? Here's the circuit diagram, see what you think.