When you're driving, the electrical load is on the alt, though if the alt is kicking less voltage than the battery has stored, the battery will feed the system as well. This is why a gas-powered car can actually run for a (brief) period of time with a non-functional alt: everybody's running off the battery, at least until the battery hits low enough voltage that the coil can't do its job anymore. After that, bad things happen that usually involve tow trucks.

I would be very tempted to hook up a voltage gauge somewhere (maybe hook a cigarette lighter adapter to a cheap parts store volt gauge?) and watch where your voltage is at when the head unit shuts down.

I know on my W124, if it's cold out, the car is freshly-started with glow plugs having just run, the lights, fan, defroster, amplifiers, and air ride compressor are all running, sometimes my HU will shut down for a minute, which I'm 99% sure is a low-voltage condition. Time to invest in a bigger alternator for me.

EDIT: Stop! Grammar time!