As a followup, in case it can help anyone else that runs into a similar issue:
It appears that the problem was with Polisy not talking to either of my serial PLMs. The PLM I was using when the problem first appeared was of the newer version, so I tried using an older (re-capped) PLM to see if the symptoms varied. They did not. Neither PLM would communicate with the Polisy.
So the next step was to check the cable from the PLM to Polisy. There's little that can go wrong with the serial cable, and the cable passed continuity checking on all connected pins.
Finally, I tried connecting each PLM to my old ISY-994, where each worked correctly.
So... it would seem that the Polisy serial connection for the PLMs had failed. There is a very slim chance that some oddness with the serial cable I used (and had been using for months) could have some kind of glitch that a simple continuity check wouldn't have detected.
I chose to replace the Polisy with Eisy, and also to purchase a new USB PLM. Everything now works.
