Regarding using your small generator to charge things aorund the trailer.
This is the front panel of my 2kW Honda EU2000i generator. It has a 8A DC socket as you can see to be used for charging things but keep in mind this will not do much for your DEEP CYCLE BATTERIES. My DC VOLTS coming out of this connection is around 12VDC but is not regulated. It varies up and down it appears...

8A is not much of charging source. With only 12VDC at 8AMPS capacity I suspect it would take around 24 hours or more to get your 12VDC deep cycle battery back up to its 90% charge state. It is not a good idea to just keep doing 30minutes or so charging of the deep cycle battery and keep using it. It will eventually go south on you.
Battery science tells us if you want to recharge a 50% depleted deep cycle 12V battery you will need to hit it with 14.4VDC with the capacity of 17-20AMPS of DC CURRENT. Doing this will re-charge the deep cycle battery from the 50% charge state to its 90% charge state. In my case my batteries at 90% charge state will provide almost full performance as if they were charged to a full 100% charge state. However I will start doing damage to my batteries if I just keep doing a 50% to 90% charge function and not do a full 100% charge. The book says to do this after 12-14 cycles of the 50-90% charge states.
Using the above listed charging setup of 14.4VDC with 17-20AMPS capacity this will take around 12 hours to get the 12VDC deep cycle battery to its 100% charge state.
Knowing what DC VOLTAGE and DC CURRENT you need you can see the 12VDC @8AMPS is not what you need and you can only get these charging levels from your ON-BOARD CONVERTER/CHARGER unit.
This is why I don't use the 8A source for anything except maybe a portable lamp to see things with when running m generator. Its best to plug the trailer shore power cable directly to the 120VAC generator receptacle using a RV30A-15A adapter from WALMART..

What I did to solve my low wattage 120VAC emergency power requirements was to install a 300WATT PSW INVERTER and run a couple of extension cords to one going to the home entertainment area and the other going to the bedroom table are. I eventually went to a 600WATT PSW Inverter to use since I found out how neat it worked for the small things when camping off the power grid...
Then once a day i re-charge my battery bank using the 2KW Honda Generator starting at 8Am each morning to get my batteries from their 50% charge state to their 90% charge state.
This is what works for me when camping off the power grid...
Just passing along some of my experiences with this... Everyone does it differently so stick with what works for you I reckon...
Roy Ken