I'd definitely check the battery - it sounds like the car is running off the alternator when the engine is running and there's very little "buffer" left in the battery to abosorb surges in demand...
if it helps, my OEM BMW battery - the largest 750 size - died overnight last month. One cell was toast (wouldn't charge, boiled on starting load) and that was the whole battery gone... behaviour was similar, except it couln't turn over the engine - everything else went, and after jumpstarting the instruments brightness would surge with engine revs, too. New battery, now no problems.
HTH
Max