Here's a nice easy fix that I wasn't predicting would be this easy.
As well as a 236 (which works perfectly), I've also had a Keithley 238 for a year or two, and ever since I've had it it's never remembered settings over power cycles.
It wasn't a big enough deal for me to want to open it up, but in the meantime, it developed a new fault which has become increasingly common where it would just shut down after a few minutes for no apparent reason. The display goes blank, and the only way to resolve is to cycle the power.
Eventually when I had some time a few months ago I checked the Lithium coin cell backup, but it had a reasonable voltage on it (3.3v). I replaced it with a new one anyway. I also looked over the caps on the power supply and all seemed good, so I just thought I'd put up with it, and leave it in the too hard / rainy day pile.
Today I found time to take took another look,with the intention of digging about a bit more.
First thing I looked at was the Lithium cell again, and again it had a ressoanble 3.3v on it. I then looked at the CMOS RAM chip and checked its Vdd, but there was nothing. There's a DS1210 chip as a battery backup supervisor, and neither battery input had anything on it.
It turns out the clip on the coin cell holder was only making occasional connection. I cleaned it up and bent the clip it a little to encourage it, and sure enough it's been working fine for several hours now without any shutdowns, and of course it now remembers the settings after I restart it.