brad - Your converter/charger unit is working just fine otherwise you would not have 12VDC items working when you are on shore power. Both converter/charger and the battery feed the same 12VDC Distribution Panel where your 12VDC fuses are located.
It seems you have a disconnect between the 12VDC Distribution Panel and the Battery only. In some trailers this is a in-line fuse located close to the battery terminal. Other trailers may have an inline circuit breaker here.
Your landing gear is connected directly to your batteries terminals but is seems the 12VDC Distribution Panel is NOT.
With the Shore Power ON you should read 13.6VDC on the battery terminals indicating that all connection between the distribution panel and the battery is GOOD. The reverse of this is without the shore power ON your ceiling lights should operate from your charged up battery. You should be able to read 12.6-7VDC on the charged up battery terminals when the shore power is OFF. You also read this same 12.6-7VDC on the large BAT cable feeding the 12VDC Distribution Panel when the shore power is off.
I suspect you have a bad connection or a blown fuse or perhaps a bad inline circuit breaker or maybe a simple battery disconnect switch opened (OFF).. Something is not connected between the 12VDC Distrbution Panel and the Battery Terminals.
Continued operations in the mode will eventually end up with a discharged battery and if you leave a battery in a 50% or lower charge state for a long period of time you may do damage to your deep cycle battery.
just my thoughts
Roy Ken