I'm not sure I understand your issue, but sounds like you need to install a 12V connection in the living room for the inverter. I use a small 300 watt inverter when we are boondocking.
If the cable is already in the living room you'll have to go into the bedroom to turn on the antenna amp.
I have seen other RVs without 12V outlets near the TV antenna connection. Many only have the antenna power plate 12V connection.
I don't have the TV power issue, but have added several 12V outlets in our campers so we had places to charge our phones.
It's a simple task, only 2 wires, you just have to decide where a good location is to put the 12V recepticle, and then run a couple of wires to it from an existing circuit. You could run all the way to the fuse box if you can't find a circuit fused to carry the TV/inverter load. I used 14ga. stranded wire so it would more than carry the power required for the inverter and TV.
