The battery will not effect the shore power connection. However, virtually everything in the coach is 12 volt
. Except for the 120 volt outlets and the air conditioner.
If the house battery reads 12.8 volts it is fully charged. The fact that all systems work when you are connected to the TV leads me to believe there is a problem between the battery and the DC panel or there is a problem with the negative side of the system where the battery connection is made, either at the battery or at the frame.
What is the voltage out of the converter when connected to shore power? Are you sure the converter has the needed 120 volts to operate?