Well you are not a newbie, so what to do? You do need to have the inverter on better wiring to that one battery in use.
If you put the inverter (that has wires with clamps) outside on the battery you can:
A. run an extension cord from it back into the trailer via an open window or door and plug your TV into that.
Which lets all the mosquitoes in. Or
B. You can plug the shore cable into the inverter out there (converter off, fridge and WH on gas not "auto") which means no mosquitoes, and your receptacles are live to plug the TV etc into,
but you can't turn the inverter on and off when not in use, which might matter in extreme conservation situations, but really isn't much of a problem to leave it on overnight and shut it off next morning. (assumes it has no remote on/off to inside the rig). Or
C. You can clamp it to the battery lugs on the DC panel inside and run short extension cords /power bars to plug the TV into. Now you get no mosquitoes, but you do keep tripping over the wires all over the floor, whenever you get up for more popcorn or beer, maybe disconnecting it all at a critical point in the movie (much swearing) and you can hear the horrible little fan on the inverter whenever it comes on. (BTDT--once! :))
