I would first change the battery. It seems like the battery is part of your problem. Also, this is an easier and least costly procedure if you're just working on trial-and-error.
This might clear up your problem. I would think that the Battery light would come on if there was a problem with the alternator, but I wouldn't rely on it 100%. Just change the battery first and see what happens. How old is the one you're currently using?
2006 E350 w/ 155k miles (Daily Driver)
1993 300E 3.2L Sedan w/ close to about 300k miles
2003 E500 Brilliant Silver (Had 217k miles when totalled!)
1989 300E with 289,000 miles (had for <1 yr while in HI)
03 CLK 500 cabrio (Mom's)
2006 C230k (Dad's)
1999 S420 (Mom's/Dad's)
2000 C230k Sport sedans
2001 CLK320 Cabrio (Mom's)
1995 C280 My First Mercedes-Benz... (155k miles. EXCEPTIONAL AUTOMOBILE. Was Very hard to let go of!)