The best way for me was to install a 300WATT Inverter as close to the battery bank as you can get and still be inside the trailer. Then run a 120VAC drop extension cord to where you want to use it for your HDTV and other home entertainment as well as camping off the power grid items.
You also may want to run a second 120VAC drop cord to the bedroom night stand area. All of these 120VAC extension cords can be completely out of sight. All that will see is the multiple receptacle 120VAC extension cord head sitting in the back of the cabinet or table.
The Inverter I have is remote controlled and I got the 300WATT PURE SINE WAVE VERSION first and then kept wanting to add a few other nice things to use when camping off the power grid. I finally ended up with a 600WATT PSW AIMS BRAND Inverter ($149-from The Inverter Store)
Has been working fine for us for the past four years now. Of course all INVERTERs are battery hungary and you may want to at least install a second 12VDC Battery or better yet get two 6VDC batteries and connect in series for 12VDC. Successful camping off the power grid works great for us with around 220AHs bank bank capacity.
This is a simple drawing showing how you can install a PSW Inverter in the pass-through storage area as close to the battery bank connections as you can and still be inside out of the wet weather conditions for the Inverter...

I am not a fan of using the 12VDC cigarette type plus for power connections just for the same reason you are now having trouble with. Direct connection to the battery circuit is the best way to go IMO.
I am also not a fan of using MSW type Inverters especially when electronic devices are in the mix. Others get away with the Modified Sine Wave type Inverters but eventually someone will plugin something since the extension cords are laying there in the open and it will go up in BLUE SMOKE. I have lost one electric blanket controller and a couple of 120VAC box fans from using MSW type inverters. Anything using an transformer adapter will get hot on you and TVs and audio items will not work like you want them to with strange noises in the audio or funny looking lines running across the TV screen. To me it just not a good idea however others will tell you it works just fine for them - so there you go... Everything you plug into a PURE SINE WAVE Inverter will work the same as being plugged into a regular shore power connection.
The small extra cost these days getting the PURE SINE WAVE type inverters to me is well worth the difference and peace of mind it will always work.
Just me yaking....
Roy Ken