The alternator has been replaced. OK. Is it actually putting out the rated amperage? Because your last clue makes me think it's marginal at best. Turning on the lights and/or air conditioner should have very little if any effect on engine speed and smoothness. I think you're running on battery mostly, and it's barely able to keep the ignition running.
Quotes to live by:
guessing gets expensive
nothing is more expensive than a cheap BMW
Ken Kanne, Silverhill, AL, Honorary Forum Grandpa/Craigslist addict/Hoarder of all sorts of stuff
1995 318is "Bebe"; 1993 325is "Elvira" 1985 635CSi "Katja" 1984 633CSi "Sylvia"
I NEED A NAP, DANG IT!