Most (if not all) commercial aircraft should take external power without the battery being placed on first. The first question..... do you have a "External Power Available" light?
The Bus Protection Panel has a internal transformer rectifier. The output of the T/R (28VDC) should turn on the light if the ground Power Unit (GPU) is outputting the correct voltage and phasing.The path to the coils on the external power contactors must be complete. This path includes pins "E and F" of the external power plug. The contacts of the external power switch. Finally, the "trip" contacts of the APU and generator contactors.
In your situation, it looks like the aircraft is using an alternate source of 28VDC from the battery bus (shown in yellow on left of image) into the Bus Protection Panel. All the other wiring must be correct to have the aircraft take power "at all" with the battery on or off.
I'd be trying a replacement Bus Protection Unit first.
The image above is quite poor after mark-ups. The original can be found