Does it have to be flex? Coax I would think would be more common for cryo / vacuum applications. Not sure where you go shopping for physics lab toys, but such a supplier would be the first place to check I think. It's going to be $$$ either way.
I think I'd be pretty surprised if flex can be made in the same way as usual, from metals or alloys other than fairly pure copper. The ductility and electroplatability of elemental copper is perfectly suited to its use in printing, which is a wonderful coincidence for being the second most conductive metal. One option would be vaporized or sputtered metal (thin films), which can still be etched like copper circuits, and is suitable for many elements, alloys, and even nonmetallic compounds, but the deposition process is very different and probably expensive, especially if you need a thick layer.
For any connection at all, you might settle for thermocouple wire, which of course is available in various special alloys... typically in matched pairs (though you probably only want just the single strand..). You might well be better off making your own from wire strips and Kapton tape.
Why is the resistance a problem? Is it not simply sufficient to put a 1 ohm resistor in series, so that that 0.1 to 0.001 ohm variation from the cable disappears in the same way? You literally gain nothing by going to the trouble of getting better (or worse, really) alloys. In the extreme case, a current loop has no regard whatsoever for the resistance, which is why it's been so popular in low-rate signaling for the last.. century and a half or so.
Likewise, for the transmission line, if it's normally losing 0.5dB, and it loses 0.05dB when cold... why do you really care at that point?
Anything you can do to arrange the measurement as a ratio, or with Kelvin leads, or a locally matched difference, or so on, will not only improve such problems as tempco and noise, but will only improve the measurement of whatever physical phenomenon you're trying to measure.