Fuel pump relay supplies power to charge the system.
After that happens the oil pressure sending unit provides power to the pump.
Fuel pump relay is normally mounted under the dog house on the driver's side.
The oil pressure sending unit is located along the bottom of the block near the power steering pump.
I would get a fuel pump relay.
However, if the rig fires with starting fluid and then shuts down, I would say the oil pressure sending is the culprit.
The fuel pump relay stays energized for a few seconds powered off power circuit to oil pressure sending unit.