If you use a 12 VDC test light you might be able to find a blank switched fuse in the fuse box and then wire into it. I am planning to something similar on my coach and my experience with Ford is there is always a spare location or two in the fuse block. Normally there is at least one switched and one unswitched.
Have you looked for a 110 outlet behind your fireplace? My coach has a 110 outlet (not tied to the inverter) in the cavity behind the fireplace which the fireplace is plugged into. I have to remove an access panel to get to it. You might be able to relocate that box for better access.
Looking at your floor plan I believe you should have at lease 2 100 outlets in the area. One should run off the inverter for the TV and the other should not come through the inverter for the fireplace.
The weekend I was on the hidden treasure hunt armed with a cross tip screwdriver and zip ties, I also found that behind my fireplace is the splitter for the cable/antenna cables. Since they were just dangling in the breeze hitting against anything in their path, I took out a trusty zip tie and secured them to a cabinet support.
FWIW I am considering installation of a pop up power station on my dining area countertop and plugging it into the outlet for the outside TV which is hidden behind an access panel in the cabinets below. Although I have outlets on the front of those cabinets I don't like wires running across my legs when I am sitting in the chair. This is what I am considering installing.
Us Plug Aluminium Alloy Shell Tensile Type Pop up Power Desktop Kitchen Socket Kitchen Plug Accept Customization - - Amazon.com