You need to post the model of your converter/charger. Some of the older units use the battery as part of its DC REGULATION CIRCUIT and has a RELAY that is operated by a working battery before it will OUTPUT DC VOLTS.
The AC and DC DISTRIBUTION PANELS is the heart of the Electrical requirements. The 120VAC Portion of the Distribution Panel is tied into the SHOE POWER or Generator source. The 12VDC Side of the Distribution Panel is connected to both the 12VDC sources coming from the on-board Converter/Charger unit and the BATTERY. Both are connected to the same point on the 12VDC Distribution panel. This allows the 12VDC Distribution Panel to be powered up when your are connected to ore power or just connected to the battery when shore power is not available.
Here is a simplified diagram of the 120VAC and 12VDC items connected to the two Distribution Panels..
From your description of problems it is telling you the Converter/Charger unit is not providing 12VDC to the 12VDC Distribution Panel for what ever reason. Things to check is the 120VAC circuit breaker feeding the Converter/charger unit, DC OUTPUT Voltage coming from the Converter/charger unit that is connected to the 12VDC Distribution Panel. It appears the Battery connection are OK since the 12VDC items work when the Battery is charged up. When on Shore Power the converter/charger unit is what is suppose to charge the battery.
You will need to get a inexpensive MULTIMETER from WALMART-LOWES-AMAZON to know for sure what you are reading...
One quick test is to measure the battery terminals without shore power turned on and your battery should read 12.6-7VDC if fully charged. The turn on shore power and the battery terminals should jump up to 13.6VDC telling you the converter/charger unit is working. This is apparently not working from your description.
You will need to find out the status of the converter/charger unit. If it does turn out to be a bad converter/charger unit do not replace with the same 2006 vintage unit. This is great time to upgrade a smart mode multi-voltage model converter/charger that should fit in the same bottom compartment.
just some of my thoughts here...