Thanks all for the input. I begin to realize that this is a more niche application than I expected. And sure, if it were for a national lab I'd be on the phone with Belden and get a quote ...
The Keithley 181 isn't all that sensitive though (specified for 30 count pk-pk noise within 2m on the 2mV range).
Wrapping a shield on a twisted pair cable myself didn't cross my mind, I must admit. Meanwhile I considered cutting the isolation of previously mentioned Cat7 cable and slipping that mesh off it, but I doubt that is practical. Instead I just used the Cat7 as is (after cutting its connectors), reusing the Amphenol connector I had (*), i.e. soldering two pairs (one conductor of each) to its soldering cups. On the other side I soldered short (~1/2") strips of #20 bare copper to ease using the barrier strip terminals.
I'm aware that those are now numerous unlike metal transitions, but the original Keithley connector with the copper pins is long unavailable and I have neither welding- nor crimping tool (and I can't crimp w/o proper tools, I tried). But since those are more or less symmetrical and in like temperature zones, it shouldn't be
too bad.
It turned out actually better than I hoped:
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The V181zero graph is the voltage as reported by the Keithley 181 across a short (~ 1/2") bare copper strip shortening one of the inputs of the 7064 relays card sampled once every 30s. There is quite an offset (~1 uV), which took a long time to reach (I didn't watch the clock, but it might have been 2 or 3h), but eventually became pretty stable (here I used the 20mV range in a test, earlier I used the 2mV range and the noise was within specifications (for the shortening plug!)) (as is the offset for the 7064 relays). One can see a temperature dependence (mostly a dependence on hot air from the a/c blowing on the Amphenol connector I'd think), but it's less than the thermal EMF fear mongers made me think it would.
(perhaps noteworthy is that currently ambient temperature is rather cool for California and that both, the Keithley 181 and the 405 switch get rather warm and so does the attached Amphenol connector)
For the measurement of the difference between the voltage of the HP735A and the LT1021 VR (divided to ~1V), i.e. what I actually intended to do, I just sample it twice: once normal and once with reverse polarity. That gets rid of the thermal EMFs (and also the offset caused by the significant bias current of the 181 across the 1kOhm output resistance of the 735A), at least those between scanner and 181, not those between source and scanner of course, but that's a challenge for another day ...
(*) I dismissed the earlier idea of connecting to the amplifier directly. I wasn't quite sure how to route the cable into the chassis (fuse socket came to mind) and prevent mechanical tension.
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