You are the second recent person that is trying to use an inverter as a whole house inverter. It may be that the inverter bonds the neutral and ground and when you plug in the shore cord both neutral and ground are connected to the battery negative post and to the inverter negative input and the inverter does not like that setup.
The correct wiring setup is to put a transfer switch or DPDT switch with inputs from the inverter, your CB output for the plug circuit and the switch output to the remaining plugs.