This problem is solved at least for the time being, but I thought I’d document it here so maybe others with the same problem can try this very simple solution.
Yesterday, after two weeks of working just fine, my wife’s FP3+ suddenly rebooted while she was using it. The boot process failed with strange blue patterns on the screen and the phone ended up in recovery mode with the options “Try again” and “Factory reset”. After selecting “Try again” the phone booted successfully, but the fingerprint reader was no longer recognized. The entries in the settings referring to it were simply missing. Touching it (screen off or lock screen) did nothing at all. Another reboot didn’t help.
I then shut the phone down, removed the battery and the SIM card, and tightened all the screws (some seemd a bit loose, particularly the one next to the fingerprint reader). After turning the phone on again, everything was back to normal and the fingerprint reader now works just as semi-reliably as before. Fingers crossed.