Hi Volkemon,
Good idea moving this topic into it's own space.
I think your 2006 is identical to our 2007 with these model year deviations.
1) You have a single house battery located in the area under the coat closet. That space is where our 2000 watt house inverter is. Our two house batteries are below the floor, between the main entry door and rear tires. Like yours, we have access from an exterior door.
2) You do not have a house inverter.
3) Your kitchen galley has a flip-up counter over the right side drawers. Before reconfiguration, we had a cabinet door covering a slide-out butcher block cutting board and two drawers.
4) Your Ford dash board is a medium gray color, our 2007 dashboard is tan.
My point here is that your rig and our rig are extremely similar, so if you have some model year specific questions, I think I can help you fairly well.
So now to your lack of 12V power.
I believe you have the same florescent light fixtures we had. They are 9-watt light fixtures which resonate and flicker when your battery is not fully charged. As I did
HERE you can cut your lighting power usage by 50% by converting the fixtures from florescent to LED in a very affordable manner. Not just affordable, you reuse your original fixtures. Replacing your fixtures will leave imprints and holes. I advise to use every-other tri-LED circuit (12 circuits per fixture) which gets you to 50% power usage. Your light-finish maple interior needs less light than our dark-finish cherry.
I also replaced all incandescent bulbs to LED for a greater reduction in power consumption, also covered in that thread.
Converting to 100% LED helped us with two house batteries. You will notice the benefit even more.