I took my phone into an Authorised Service Provider, who ran a diagnostic and said there was nothing wrong, but that the symptoms I described matched up with a software issue that had been acknowledged by Apple. Apple’s documentation on the issue, they said, indicates that resetting the phone to factory defaults will resolve the issue, but if you then restore a backup affected by the issue, the problem returns. The Authorised Service Provider advised me my best bet was to either wait for Apple to fix the issue, or take my phone into an Apple Store.
After waiting a few more weeks, I took my phone into an Apple Store. The Apple Genius that helped me confirmed the Authorised Service Provider’s take: that the issue was an acknowledged software bug. The Genius commented that he was surprised it hadn’t been fixed yet, because Apple had acknowledged the bug quite some time ago.
The Genius offered to replace my phone under warranty, noting that it might not fix the issue, but that was all he could do for me. I accepted the replacement, and thankfully the issue has not recurred with the new phone, even though I restored my backup from my old phone to it.
My spouse, whose phone was bought at the same time, was also affected by this issue, and was also able to obtain a replacement from our local Genius Bar. The problem has not recurred for her new phone either.