I suspect you are calling your POWER CONVERTER/CHARGER unit an INVERTER. These are two different animals. Most trailers come with the CONVERTER as a standard item and the INVERTER is a custom install items.
The CONVERTER takes 120VAC from the shore power hookup and converters it to 12VDC to run all of your 12VDC items and also charge your 12VDC batteries.
Provide us with model numbers and we can tell you what you have...
The INVERTER is connected to your BATTERIES and provides 120VAC to run some of your 120VAC Appliances. The INVERTERS are very battery power hungry. A 1000 WATT unit like you mentioned will draw a good 85-100AMPS DC CURRENT from your batteries when fully loaded. Thats alot of DC CURRENT to deal with.
here is a simplified 30AMP Power Configuration diagram to help you identify the different things associated with the 30AMP Power configurations... As you can see from this chart your OUTLETS only work when connected to SHORE POWER. All of your 12VDC items including the ceiling lights all work from your BATTERY or from the CONVERTER and BATTERIES when connected to SHORE POWER. Hope this helps..
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Roy Ken