I think an easy way to increase the confirmation that it is indeed the fuel pump relay is to try to start the car from a stone cold engine. If it starts and runs well until it heats up in the engine bay, then it could in fact be the FPR. A used relay should be fine, but that could have limited life...

I believe what happens in the relay is that the solder joints develop cracks. As the reply gets warmer from the engine bay heat, the cracks expand and eventually severe the connection thus terminating the fuel pump. You could buy a good used unit, fix the car, then take apart the old one, re-solder it, test it and keep as a spare in your trunk.

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2006 BMW M5 "Heidi" @ 109,000 miles
2005 MBZ C55 AMG "Lorelai" @ 165,000 miles
1991 MBZ 300E "Benzachino II" @ 165,000 miles
1990 MBZ 500SL "Shoshanna" @ 118,000 miles

(On the hunt for a good used M103 engine as of 6/10/23, PM me if you have one to sell!)