Some news from the Voltech PM3300:
I've found the culprits
#1 is that (high grade 1000uF/16V) electrolytic capacitor used to AC couple a half bridge MOSFET stage to drive all the small transformers used to supply the input modules. I found it getting warm and tested its series resistance to about 600mOhm at 40kHz, quite a bit too high. Replaced that one, but the fluctuations weren't gone and isolated supply voltages were somewhat better, but still far from good.
#2 is this pin of the power supply connector. It's the +12V supply line, carrying 1.5 ... 2.0A. Note the slight discoloration of the connector housing, and the pin itself lost most of its plating when I removed it from the housing. There's been about 0.5V ... 1V drop across this contact. After an intermediate fix, the supply voltages with OK with little to no margin, and the fluctuations are gone.
The better fix for this (crap) kind of connector would be to cut it off and solder the wires directly to the pins - maybe I'll apply this later.
Here's a view of the main circuit board (the onboard battery is still good, reading 3.6V quite as new):
Right now, I've connected 10V DC and 1A DC sources to watch the stability. The meter is specified to 0.1% (from reading) accuracy @DC (0.05% AC). Judging from the first results, these specs look quite optimistic. From my own experience I know it's quite brave to specify this kind of wide band power meter developed in the 1990s to this accuracy.