First off you are calling the CONVERTER an INVERTER. These are two different animals..
This is a typical diagram of how the 30AMP Power Distribution Center that houses the Converter is wired. May help show what is controlled by what...
The thing to remember is both the battery and the 12VDC output of the converter both feed the same point on the 12VDC Distribution center. This allows the battery to operate all of the 12VDC items when you are not connected to shore power.
The 120VAC items are controlled by circuit breakers and only works when you are connected to shore power. The 12VDC items are controlled with plug-in fuses and works from both the CONVERTER and the Battery connections...
The best way to tell if your battery is being charged is to use a hand held multimeter across the battery terminals. If the battery is fully charged and you are not connected to shore power you should read 12.6-7VDC. Plugging into shore power you will see the DC voltage measured at the battery terminals jump up to 13.6VDC telling you the converter is charging your battery.