Thank you guys for replies !
I managed to cut and rearrange some traces on the UT61E+, related to current passing and voltage drop sensing.
As a result: before the mod, with 6.000A stable current (I could guarantee its stable over time), I had 13-14 mA sensing increase after 20 minutes. First 7mA in first 2 minutes, following 7mA - later during 18 minutes. I did not see a sense to track it longer.
After my mod: with the same 6.000A current source, I had maybe 1-2mA increase over the same 20 minutes.
With a 10A current source (which was less confident before the mod), I get 2-3mA increase over 1 hour period! This also includes an addition hack to avoid the 10A fuse usage and supplying current to the PCB without original socket contacts.
Yes, I know you may not like it, but my goal was to track high current stability in a circuit for longish time.
And my UT61E+ now does not overheat and I do not get the "Cut!" alert and stopped current reading.
Here is an offender on PCB - cooper track with solder, on left from COM label!:
even just removing a bit of solder, current reading has increased by 50mA on the 6.0A flow:
Note - this is a place where input jack plate soldered too. Imagine if you needed to resolder it for a reason. You cannot do it without breaking the join resistance, which actually is "a part" of shunt !!!
Woahhh, ... I needed to try and test then with the shunt position and after a few attempts I was lucky to catch -2mA reading on the 6.000A flow.
Note - before the mod it was -7mA reading from cold state. So accuracy became even better!
I knew it was a risky attempt, but I did similar things in the past and I knew how it should be made.
I'll try to record a video what I did.