You might want to check a switch on the dash. I have an '87 300E that gave me the same kind of problems. Turns out that the switch for the rear light, not the dome, was on. You don't notice it unless you drive at night. My wife had dusted the instrument panel and accidentally turned it on. It is enough draw to run the battery down over night.
Anziani

'87 300E 201K

'95 E420 110K