Won't the 400w do all that? If it does not because it is limited by being plugged into a 12v socket like on a Winegard TV panel, then just switch to the clamps set of wires for the inverter instead of the 12v plug set, and clamp to the battery lugs on the DC fuse panel.
That gets you the power, but now you trip over it going to the fridge, so now add some #8 wire to the lugs and clamp to those and keep it all off the floor.
time2roll managed to get his inverter behind the converter and out of sight entirely doing it that way.
Use a power bar with its own on/off switch plugged into the inverter to plug the TV into. Turning off the TV still leaves it drawing some power in standby mode.