Glove Box Light?
I had a similar problem with my 1986 BMW 635csi. It turned out that the glove box light wasn't going out when I closed it.
I don't know if your glove box light is on the same circuit that's drawing heavy current (my 95 E420 doesn't have a glove box - so I'm not even sure if there's a light in there). But check this out as a possibility.
troy 1995 E420 70k