I am not sure what your SKYLARK floor plan looks like but I assume it has the V front. Probably similar to this 2012 SKYLARK floorplan.
What a lot of folks will do is to get 300WATT to 600WATT PURE SINE WAVE INVERTER that has remote ON/OFF and mount in the storage bay close to your battery install. It should not be installed in same compartment as the battery but needs to be probably 5-feet or less away from it. The 12VDC cables from the INVERTER should be directly connected to the battery switch along with its dedicated covered FUSE BLOCK.
You might want to consider this typical floorplan of a trailer your size with the idea the INVERTER will be installed somewhat like shown here and then run two typical extension cords from the INVERTER to maybe the Home Entertainment center area and the bedside area. The length of the 120VAC extension cord is not an issue like the 12VDC cable is. Use an extension cord that has multiple 120VAC sockets and a separate ON/OFF switch similar to this one shown here. We plug in the home entertainment items, cell phone chargers, computer charger, NOAH WX alert radio, fan, desk lamp, motion detector items, etc. The emergency use items....
As stated by the other post the use of INVERTERs require more batteries capacity as they will load down ONE battery rather quick. What ever you pull out of the battery you have to put back in but since you have 2012 year trailer with a "three stage smart mode converter/charger" on board along with the 2KW Honda Generator you already have it will be real easy for you to keep your batteries charged up once a day with a three hour generator run time. Most places you go to camping will allow you to use your generator for this short time period during the daytime only.
I am not familar with the 2012 SKYLARK with the V nose but suspect you are limited on battery space. You would want to install two of the 6V GC2 gulf cart batteries from COSTCO or SAMs CLUB to give you a 12VDC Battery Bank system of around 200AHs with the plan in mind to connect your trailer shore power cable to your 2KW Honda generator at 8AM each morning when camping off the power grid to allow the on-board smart mode converter/charger system re-charge you batteries backup to their 90% charge state so you can do this all over again the next day/night camping run.
This is how we do it here with a 255AH battery bank and are very successful camping off the power grid up to a week or so and can run everything we normally use at regular electric camp sites from the 120VAC and 12VDC appliances except the air conditioner and the high wattage microwave. I will draw around 20AMPS of 12VDC from my battery bank between the 8PM to 11PM time slot and then around 1 AMP thru the rest of the night until 8AM the next morning when I am allowed to run the 2KW Honda generator to re-charge the battey bank. We watch HDTV, plays DVDs, I play with my ham radio some, momabear uses an electric blanket on her lap if it is cool, run all the interior LED lights we need, run all CFL exterior lights we need.
Note in my floorplan (tent beds not shown) I have two locations where a multi-purpose 120VAC extension cord is secured to the back of a cabinet top location which provides me with two 120VAC "Emergency Power" drops from my INVERTER. This is the only part visible as the cords are hid by the use of the wiremode products (LOWES) . I use a 600W PSW INVERTER now for the two 120VAC drop cords shown here. Your 300W-400W should do just fine. My PSW INVERTERS are AIMS from THE INVERTER STORE.
I would definitely use the pure sign wave INVERTER as this will give you peace of mind that nothing is going to go up in blue smoke. Using the MSW type Inverter will probably run most of your items but the PSW Inverter will run them all without burning them up. The price of the PSW type are now cheap enough to purchase as compared to just a few years back.
All of this submitted for "FOOD FOR THOUGHT" Works good for us camping off the power grid.