I had a very similar no-crank condition in my E300 ('98 W210 Chassis) which I found to be a bad relay. It was located under the hood, passenger side in the box near the firewall containing fuses and relays. I found it by accident essentially when one time I had the no-crank condition and then I tapped each component in there with the butt-end of a screwdriver and tried to start each time, after I hit the "right" one it started right up...then I removed the relay and it would not crank just like it was at fault.

The dealer stocked the part and it was only about $25 IIRC (a small black cube relay with 4 contacts on the bottom)...I will try and find that old post and see if I have a part number. I'm not sure if the E320 uses the same electronic controls but I think it does...mine was heat related too...it would fail when it was very hot out and after the car was warm...never failed when the car started first start of the day. It's been over a year since I replaced it and never once did I experience the same problem so it was a 100% fix.

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Marty D.

2013 C300 4Matic
1984 BMW 733i
2013 Lincoln MKz