I leave my Univolt on all the time, but I only check the batteries once per month. The older one has some trash floating in it now, but it's still got 2.2V differential between all the cells, so it's not a goner yet.
If you LIVE in the trailer, I think that the normal course of using the lights, fans, and other 12V
appliances is enough to keep the Univolt from killing the baterries. Please do be sure that you only put distilled water in your batteries. (I think some tap water contributed to my dirty battery fluid, but that was before I owned it, and it came from the frigid north, so I'm not sure what effect extreme cold has on batteries.)
I'm curious -- where and how are you measuring the energy usage?
I'm sure somebody with more knowledge will chip in here, but I think you need to keep your batteries in the circuit. They "smooth out" the power from the Univolt. I know from my Argosy that if you run the Univolt without a battery, you end up replacing a LOT of light bulbs.
Hey, you have two brand new batteries? Great. I'd be trying to take care of them and prolong their life as long as possible.