I've seen more than a few in-tank electric fuel pump wiring issues,often the wire(s) from the fuse box back to the tank get all gangrene in one spot,or at the plug at the tank (most new pumps now come with a new plug)...the wire conducts enough amps to power the pump up briefly ,then heats up and the connection breaks and the pump quits--re-starting it sends enough of a voltage surge to run the pump again for a few seconds..
My friend had quite a few GM's in his shop for this issue and he used to just put a new fuel pump in assuming it was dying (they can create the same symptoms your having)--but after he was done installing them the vehicle would either have troubles again shortly after the customer got it back,or right away..
He learned using his digital volt meter to check for power at the pump wiring was not a good indicator of the condition of the wires--a wire with lots of corrosion or just a few strands still making contact can show a healthy 12+ volts on a meter,but fail to deliver enough amps to power the pump right..
He now uses a halogen sealed beam headlamp for a tester--if the wires can power the bulb for several minutes with no dimming or flickering ,then they are probably OK--I watched him test a Buick this way he had put a new pump AND the plug on the harness on,it stalled before he backed it out of the shop..
He asked me to get in the car,put it up on the lift,and had me turn the key on after he hooked the headlamp bulb up to the pump's hot wire and ground--it took about 2-3 minutes before it started dimming,then went out completely,and running his hand along the wire,he felt a spot that was hot and "mooshy",he cut it open and there was only a few strands left making contact,all green and powdered..
That car also had a bunch of ground wires for the ECM ,that was under the right side kick panel,all plugged onto a "bus bar" that was bolted to the sheet metal,that were all green and corroded--he cut them all back and stripped them bare and soldered a ring connector to all of them and put a new sheet metal screw to secure them to the body panel,after grinding the paint off to bare steel..that car had been in numerous other garages around town,and he was the only guy who "fixed" it for good!..
And he now tests the wires FIRST,before wasting a few hundred bucks on a new pump that may not be needed too!..(the pump he removed was in the car only a month or two and was fine,but the customer wanted another new one regardless while the tank was out--he also put in a new tank & sending unit as both were getting real crusty..)