I simply checked the noise on your 181A with the 3458A, in its 10mV A.C. range.
This did the job very well , if I remember right, it was about 0.5 mV rms, at the 1000x amplifier output
Frank
P.S.: That measurement was the 'pump-out', as Keithley used to call it, it's nothing else but the chopper spike suppression, which has to be trimmed.
These 520µV, measured easily with the 3458A, were well within the limits, but that has nothing to do with the amplifier noise, and also a pre-amplifier is not necessary for fault-finding.
The error is definitely located in the pre-amp, as the 2V .. 1kV ranges were ok.
What's really strange is that the excessive and sporadic noise, of 500nV up to several mV is in the 2mV range only (resolves 1nV or 6 digits), but it's not visible at all in the 20mV range.
As the 20mV resolves 10nV, it should be visible on the 2nd last digit, but there's nothing.
And the only difference in the setting is a reed relay, which is switched off (inactive) for the 2mV / 1000x range of the pre-amp.
Therefore, all semiconductors might probably be ok.
Either this reed relay is not ok ( so de-solder it in first place).
Or the whole pre-amp might oscillate at 1000x, which I didn't check with an oscilloscope, only with my 34465As digitizing function, i.e. at 50kHz sampling rate / bandwidth.