Had to replace my fuel pump a few years ago - situation different - warm/hot weather.

When it warmed up - temps over 80 degress - car would shut down after 15 miles of driving and that was after starting out cold in the A.M.

Let the car sit for an hr. or so and it would restart and drive for 3-5 miles B4 dying again. Let it side overnight and again - 15 miles B4 croaking

Not saying fuel pumps only die in warm weather - just making a comment.

I had a situation similar to the OPs - turned out to be the OVP.

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Mike Murrell
1991 300-SEL - Model 126
M103 - SOHC
"Fräulein"