Update to yesterday's post on the "dead" i3:
Charged battery for about three hours on a 6A smart charger. Reinstalled, and the car came back to life as if NOTHING had ever happened. I did have to set the time, but not the date.
However, I am pretty knowledgeable about 12V electrical systems and batteries, having worked at worked at West Marine for 30 years, been a member of marine tech boards, etc. Before I reinstalled the battery, I did a minor "load test" on it to determine if the battery was truly charged or whether it had a bad cell or overall deterioration. I connected a large bilge pump to the off-charge battery for about 30s and monitored the voltage. It dropped quickly to about 12.5V, but stayed there as the bilge pump continued to run, drawing about 7A. I know this isn't a complete capacity test, but a battery with a bad cell would show a pretty decided decline in my opinion. Ideally, I'd put a larger load on it, but it's only a 20Ah battery. I used to have a Reserve Minute test device which was totally cool where you could adjust the load on the battery and the device would turn off automatically at 1.75V per cell.
Bottom line: I think the battery HAS deteriorated enough to cause the car to go whacky when it's not left on the 110V charger, and that a new battery is the correct solution. The only question I have is whether I should also take it to the dealer to have them see if anything else is wrong.
Chuck