OK, power for the fuel pump is supplied by one of the relays in that "dual relay" (located next to the injector resistor pack, typically) during the engine start and by a set of contacts in the air meter once engine has fired and there is continuous air flow through the air meter. Your indications are that there's nothing wrong with your "dual relay" but that the problem may be with the air flow meter contacts instead.
A quick test is to remove the air filter lid with the air meter and push the air meter vane/flap gently with a screw driver while the ignition key is "ON" . . . the fuel pump should run as soon as the flap is moved slightly off the "rest" position.
Here's detailed
Bosch L-jetronic FI HTML