You won't have much luck charging your battery off the BAT CHG plug of the Generator. This only gives around 8AMPs of DC current and is an unregulated source of 12VDC. There is only enough power here to give you a good trickle charge to your trailer battery. You will get behind the charge curve in a hurry...
You best way to charge your battery is connect the trailer 30AMP Shore power cable directly to the generator 120VAC receptacle using a RV30A-15A long 'DOGBONE' type adapter. You can get one of these from WALMART very cheap.
Your 2KW Generator will operate the on-board converter/charger and if it is a smart mode charger it will re-charge your trailer battery in a three hour generator run time.
If you do not have a smart mode converter unit then I would think your best thing to would be pick up a smart mode converter/charger unit for the trailer. You can also just pick up a smart mode portable battery charger like the B&D VEC1093DBD 40AMP charger and plug that into your generator and it will re-charge your batteries to their 90% charge state in a three hour time frame.
We camp almost entirely off the power grid in our OFF-ROAD POPUP and this is our game plan. We have a smart mode PD9260C converter/charger unit, added batteries, bigger battery cables, changed out ceiling lights to LED lights and this allows us to do almost anything we do at electric sites except run the air conditioner. We have it planned out to run what 120VAC items we want to run from an 600WATT PSW Inverter and 12VDC items from direct connected to the battery and can make it through one day/night run off our 255AHs capacity battery bank. Then at 8AM the next morning when allowed to run our generator we will connect the shore power cables directly to the generator receptacle using the RV30A-15A Adapter and then re-charge our 255AH battery bank back up to their 90% charge state so we can do all of this all over again the day/night run off the batteries.
When the generator is running we make our grind and brew bean coffee supply for the day and can also run some other 120VAC items we want to do.
In the evening we watch HDTV, play DVD, run 120VAC fans and lights, charge computers and cell phones, run our outside patio lights and lamp post light, and like I said do almost everything we do at a regular electric campground.
It does take some planning but we have been doing style of camping off the power grid since around 2009 in our OFF-ROAD POPUP.
This is the way we do it haha...
My generator is a Honda 2KW EU20001 which sits secured in the tailgate corner of my F150 truck covered and lockable bed... When we need to run the generator we just pull it out onto the tail and fire it up... The 30AMP shore power cable is long enough to plug into the generator from there.
Only way to camp for us...