It sounds like the battery is the culprit, try the temporary swap. My batteries when fully charged sit at around 12.9 volts. If there's no parasitic draw to speak of I'd think it has to be the battery.

I turned the car off and measured the voltage across the battery again and it's reading 12.4 V. That doesn't make sense to me because there's no way the battery charged up in the short five minutes I had the car on. That seems normal because right after driving the battery has some extra voltage th...