I'm no expert on Spica, but no, I don't think the Spica pump can pull fuel from the tank and pressurize it properly. If the car continues to run with the light on, then I believe the pump is running but not developing sufficient pressure to trip the pressure sensor.
There are a couple of buzzers behind the dashes of mid-70's Alfas - one for seatbelts and the other just to annoy you. Most of these were disconnected early in the cars' lives. I can't explain what else would be buzzing back there or why your's starts buzzing when the fuel pump quits.
I'm guessing that you have two, separate problems: vapor lock causing the fuel pressure light to flash while the car is running and something electrical causing the buzzer/pump issue.