I will check that out, thanks. There's not a lot of definitive info on this topic, but I ran across some literature for Spectorcore basses (their piezo bridge assembly is similar to what's on mine) and it mentions the limitation regarding non-metallic strings there. If I just touch one of the saddles with my finger (contacting both of the tiny metal points at the string's contact point) the note suddenly rings clear. I thought maybe it was an intermittent short but after I switched out to metal strings there were no problems at all.
Edit: I should add this isn't a 'doorbell buzzer' film piezo, rather it's a complete surface mount bridge with a pickup built in to each saddle.
Another edit: looks like I may have found a better analysis of the problem, and a workable solution,
here.
If that's the case, no need for me to look for replacement strings...