Yeah... about that.
I rewrote my scripts to double-check the availability of the AP. It waits 30 seconds and checks again. The "problem" disappeared immediately.
That lasted about a month before it came back. Now, it dies about twice a week. Seven times since rewriting the script, and only once when anyone was at home to notice it (and no one did).
It's not a power failure issue, since it's on the UPS. It is using POE (Power Over Ethernet) now, so that's another change.
The next step is either triple-checking or using a dedicated wireless NIC to do the scan.