Thank you for following up with me, Kleinstein!
I agree the scope is not suitable to find differences in the µVs, and there is a lot of noise on the signal. The noise seems to be radiated or condicted EMI, not from the signal generator, which is a Rhode&Schwarz SRM RC-Generator. I get the noise on the scope even when I just connec two wires which are shorted at the far end. For this application, I do not worry about that noise too much.
The DMM self-test which fails should be just a math function, using the static values acquired before. And I really don't see what is wrong and it is still obscure which values are actually used (even though we made some educated guesses).
Sorry for abusing the forum as some kind of log book for my amateurish messing around.
I have adjusted the DC_DC turnover so that the signal at the RMS converter output is the same (within 10 µV = 5 ppm FS) for positive and negative 1V input voltage, and the test still failed.
I connected about 4.8 mV RMS to the DMM input and measured the following signals in ACV measurement mode:
CH1: TP215 Rectifier amplifier feedback path, before diodes
CH2: TP214 Rectifier input (from preamp)
CH3: TP216 isolated and inverted negative half-waves
Screenshot attached. One thing that looks a bit strange to me is that CH2 and CH3 seem to have an offset of about 3-4 mV, but I later saw that the signal generator has a small DC offset which might be the reason for that.
With no other ideas, I thought of away to adjust some of the trimmers in a defined way.
1) Offset Null R215: The rectifier's amplifier M201 gets feedback through D210 for positive half-waves and through D209 for negative half-waves at its output. Because of the diode forward voltage, there is a 400 mV voltage "jump" on every zero crossing at the input. I believe that the offset trimmer R215 should be adjusted with 0 V input voltage, so that the output at TP215 starts becoming unstable. I shorted TP214 to GND with the meter in VDC mode and this adjustment seemed to work as expected. I did only a very minor adjustment.
2) With the Offset set, I should be able to tune the DC turnover so that the rectifier (measured through the RMS converter) has the same amplification for the positive and negative half-wave. I used the pathway test functions like before and was able to adjust DC turnover so that the rectifier had identical gain for -1V and +1V. The gain is still about 1.002 from my previous tinkering, but I guess that should be OK (and it compensates the preamp's gain which is about 0.998.
If someone is trying to do the same thing - beware, the RMS converter's gain is quite temperature dependet, the value changes by approx. 100 µV when the meter is tilted by 90 degrees on its side.
At that stage, I got:
Ref mux output P059 +0.994,268 V --> Preamp out P060 -0.992,595 V --> RMS out P063 0.994,185 V
Ref mux output P061 -0.994,299 V --> Preamp out P062 +0.992,590 V --> RMS out P064 0.994,181 V
So, the preamp output and RMS converter output look very symmetrical. However, the preamp output should actually not be as symmetrical, as the Ref mux output is not as symmetrical. I suppose trimmer R111, "Bias Current" at the preamp input could fix that. Ugh, another trimmer to screwdriver into lower orbit...
After messing with R111 I get:
Ref mux output P059 +0.994,263 V --> Preamp out P060 -0.992,582 V --> RMS out P063 0.994,149 V
Ref mux output P061 -0.994,296 V --> Preamp out P062 +0.992,608 V --> RMS out P064 0.994,174 V
With the limited understanding I have of the circuit and the self-test functions, this looks very reasonable to me and I don't see anything wrong with these values.
So, I triggered the internal source calibration again, and got error 2433 again. Bummer

Yes, I could turn down the gain of the RMS converter again now. But I I think I have been there already before - anyway

Ref mux output P059 +0.994,258 V --> Preamp out P060 -0.992,581 V --> RMS out P063 +0.992,579 V
Ref mux output P061 -0.994,291 V --> Preamp out P062 +0.992,606 V --> RMS out P064 +0.992,597 V
Ok, one last internal reference characterization for today (it takes about 10 minutes each time)...
Nope, same error again.

I just don't get it...