First, 13.5V is about normal for Euro built cars (MB,BMW,Audi, etc).
US brands usually run about 14.5. Asain about 13.5. Batterys last a lot longer if charged with 13.5.
This is how I diagnose a starting/charging problem. One of these test will require a 100+ amp meter, which you probably won't have but maybe you can get some shop to check it for you.

1. Start with a fully charged battery.

2. With engine running, and no loads (no A/C, lights, etc) check the voltage at the battery. Should be about 13.5 or so.

3. Check the voltage from the back of the alternator to the battery ground. Should not be over .1 or .2 v higher than the battery voltage in step 2.

4. Turn on the A/C on high fan and the head lights.

5. Check the voltage at the battery and alternater as in step 2 and step 3. The voltage should be about the same but will be lower than with the A/C and lights off. If the voltage is more than .2 or .3 different when checking at bat and alt then there is a resistance in the wiring between the battery and alternator.

6. Now with the engine still running and the A/C and lights still on, check the voltabe between the bat neg and the alternator case. Should be less than .3 volts. If not, there is a ground problem between the bat and the alt. Also check between the bat neg and a good body ground. Again should only be .3v or less.

7. Now comes the part most people don't have the equipment to check. We need to check the amperage the alternator is supplying. Both idling and 2000 rpm. A good MB alternator will supply at least 50 or 60 amps idling with A/C and lights on and 85 or maybe a little more at 2000 rpm. Now this is the tricky part. If the voltage is low (less than 12.5) at idle and the amperage is low, either the belt is slipping or the alternator is weak. If the voltage at 2000 RPM is less than 13.5 and the amperage is low, the alternator (or regulator) is bad. If the voltage is less than 13.5 and the amperage is greater than 80 amp at 2000 then the battery is bad or there is to much amperage draw caused by something in the car.

I know all if this is very confussing, but believe me, this is how it must be done. (I do this every day. Paul's Auto Electric}
Most mechs don't know how to do this be cause of the electronic principles involved.

One thing I see the most is belts that are too lose. They must be very tight. To check, With engine on, and A/C and lights on, check the voltage at the battery. Now stop engine, tighten the belt some more and start the engine and retest. If there is any change in the voltage, the belt was, or is slipping. Believe me, I see this a lot even from other shops.

The other thing I see a lot is defective rebuilt alternators. When an alternator is rebuilt, they test them at 2000 or 2500 RPM. If they supply the proper amperage and voltage they are considered "good". They do not test at idle speeds. This gets a little complicated because of the different sizes of the pulleys on the alternators and engines. 600 rpm engine speed may be 2000 rpm at the alternator, or it may only be 1500 or any thing else for that matter.

If you can do these test and reports back, maybe I can guide you
in the proper direction. Even if you can do steps 1 thru 6 above and let me know maybe we can sort this out without you having to find some one with the amp meter.

Please excuse the poor typing and spelling. I don't claim to be a scollar.

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84 500 SEL (307,xxx miles)