For what it's worth. I had a very similar problem with my 87 560SL. I thought it was the fuel pump relay as well, based upon what I had read in various posts in the forums. It turned out to be something that has received some attention in these forums, but not as much as fuel pump issues: it was a crank position sensor. You might search for crank sensor and see what's been written. BTW, it was a local independent shop that did a thorough check to find the faulty crank sensor.

Disclaimer: I do NOT know if there are any similarities between your motor and mine! Good luck!

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Kevin Arburn
Overland Park, KS
1987 560SL