Yianni, thanks so much for posting your pictures and information. Your installation looks nice.
This is indeed timely information for me, as I'm in the process of trying to do this myself, with the addition of solar. The 12 volt battery that the dealer installed in our 3010 was pitiful. It wouldn't even start the generator in the morning after limited furnace use overnight. I have bought 4 Trojan T-105's and plan to install them in a box I built in the front compartment, but it's a tight fit. What kind of battery box do you have and where'd you get it? I couldn't find one that would fit the space and hold the batteries.
I bought a Dyno battery box
http://www.dynobattery.com/BoxGC2.pdf that is made to fit 4 6 volt batteries. You can get one at
http://www.marinenutz.com/products/Dyno-Battery-Boxes.html about $90.00
Rod(Rataryn) found this box and interestingly, he has an early 2010 and can fit this box sideways but on my later 2010 it will only fit front to back. The height is no problem and the box does fit, it's just a little tight. I installed the selector switch because I wired the dealer 12 volt in for back up. 2 trips ago I neglected to run the gen and woke up to dead batteries. Had to back the truck around and plug in at 6 in the morning to get the gen running so the kids could have heat. Got some unfriendly looks later from other campers for that one. So i just put the 12 in the mix as an emergency gen start battery.
I installed a Xantrex 1800 watt inverter with built in transfer switch. I decided to hard wire it so I wouldn't have to run an extension cord to an adapter to the outside of the trailer and have to turn off the converter breaker every time I wanted to use it. It was too much hassle so I made it simple.
Check the install manual of the inverter your thinking of buying. Xantrex does not want theirs installed in a battery compartment due to possibility of explosion from battery gas. It works well in the basement rafters and is nice having a remote/battery monitor.