I was able to link up with an engineer from Enel X (successor to AeroVironment). Since the EVSE-RS is JuiceNet enabled, he was able to read the logs, and saw a large number of PlugIn/PlugOut events. But the PlugOut events typically lasted a few hours. They might trigger the discharge behavior, but they're not recording it.
The on-off loop takes about 24 seconds. Here's what it looks like:
He thought it might be a partially inserted charging handle. The handle is solidly inserted, but that doesn't rule out a bad contact. After one dead battery a month ago, the 12v discharge has recurred three times in the past three days, which might suggest a component failure (including a connector).
Now I'm trying the charge cord that came with the car, plugged into 240v, and will give it a day or two to see if the problem recurs.