I assume you are talking about an optional small INVERTER that is connected to your battery bank. What is the part number of your small wattage 12V INVERTER
I use a 600WATT PSW AIMS INVERTER here for all of my 120VAC low wattage appliance things I want to have running when camping off the power grid.

My batteries however are charged using the on-board CONVERTER/CHARGER unit being run by my 2KW Honda Generator each morning for about three hours using the trailer 30AMP shore power cable direct connected to the generator 120VAC RECEPTACLE using a RV30A-15A Adapter (WALMART)
Didn't understand your comment about about using the inverter to charge your batteries. Maybe you was referring to CELL PHONE and COMPUTER BATTERIES...
I just looked at a SMALL INVERTER here in the Radio ROOM and I don't see a fuse or circuit breaker on it anywhere. Must have something inside the cover I reckon. Never have overloaded mine yet...
I do the same thing here with running a couple of extension cords tucked around the corners of the trailer with one extension going to the home entertainment area and the second extension cord going back to the bed area. All you see is a MULTITAP extension cord 120VAC head on the back of a table top. My MULTITAP heads do have a RESET SWITCH on them. I guess that would trip if I overloaded them with a vacuum...

Roy Ken