You can charge your battery until hell freezes over but the fact is your battery is the number one item that needs replaced. Green eye means nothing. If your battery fully charged isn't 12.6VDC or specific gravity 1.265 it isn't charged. A fully charged battery at 32F had only 65% of its CCA too.
BMWs are VERY sensitive to a battery that isn't fully charged.
Do yourself a favor before blowing all this money on the dealer, replace the battery with a new one, get the codes read by an independent BMW shop in your area and then decided. Also it is not out of the question that the door unlock solenoid is pulling more amps then it should causing issues especially since this is temperature related. However so if your battery temperature related.
#1 guess is it is your battery. What do you have to loose anyway your battery is already on life support...
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