My intuition tells me that you should simply replace the fuel pump relay. The stalling behavior mirrors my experience just before the same relay failed in my 6-cylinder. It might test out just fine for awhile (hence the car running for awhile then stalling again) and then give up the ghost (mine failed on Halloween evening with a car full of little goblins and fairies) at an inopportune time.
Perhaps others have better ideas.
2003 Dodge Durango SLT
2002 BMW E39 530i Sport
1988 Mercedes 300TE
2003 Chevrolet Suburban