It sounds to me like the 100 point or 1000 point relays need adjustment (whatever the value of the pop bumpers are). From memory, the relay picks up the switch (pop bumper skirt switch) and then the 100 point relay pulls in, closing the circuit to the pop bumper coil and the chime. That outer chime box coil is the clue.

The relay is in the head - you will hear it humming when the switch is closed. If you tap it closed with your finger (the relay), it will most likely work. There is an adjustment needed or a wire has come loose.

I wish I had the schematics to verify