I believe the OP is confusing a CONverter and an INverter.
To make is as short as possible.
A CONverter is what your shore cord plugs into, it provides the power distribution from shore 120 Volt AC power to all you outlets, as well as providint 12 volt DC to the unit as well as the charger for your battery. It ONLY works when plugged in to shore power.
An INverter is an appliance that draws power from your 12 volt battery and changes it to AC power for whatever device it is connected to.
Inverters are commonly used by manufacturers to power a refrigerator in an RV or to a television. They are drawing power from the battery which must eventually be recharged to allow further function of the appliance.
The "Loop" some have tried to create is to use the inverter drawing power from the battery, to power the Converter to recharge the power.
This loop does not work because there is always a small loss when changing the form of power so eventually it always consumes more than it generates.