Problem fixed! It turns out there was a significant buildup of corrosion on the connectors/terminals where the electrical cables from the battery terminate at the starter solenoid and alternator. My local non-BMW Bimmer auto shop just cleaned the terminals and charged me $150. It has been about 6 weeks, so I am pretty sure it is fixed. The battery always died when I went away for a week, which didn't happen when I went away for a week last month.
If you happen to live in the Toronto area, it was Bimmersport Automotive that fixed the problem (Mississauga). It's the only time I have every used them, but they seemed great.
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