SO you are going to pull the 120VAC leads that goes to the receptacle that the home entertainment items is plugged into and just connect that to the INVERTER. It will always be only connected to your INVERTER. You cannot jumper your 120VAC to this same connection along with the 120VAC that may be connected to shore power. It has to be one or the other ???
On the extension cord idea. I do not see my extension cord. It is totally hidden except where the multi-port receptacle is attached to either inside the cabinet or on the back of the table top. All of the cord per say is inside cabinets, behind walls or inside wiremold from LOWES. It just makes it easier as you don't have to pull 120VAC wiring out of the distribution panel.
I unplugged my home entertainment items and plugged into the extension cord and I just leave it that way. I am always running those items off the INVERTER.
There is another issue as well as some INVERTERS will not live well with your trailer GFCI receptacles. When you plug in it trips your GFCI circuits. My AIMS PSW 600WATT Inverter works ok with the GFCI in my trailer. This is another plus for running the extension cord idea.
These are all just my observations with my comments - just passing along what I had to deal with... Every situation is unique it seems.