I'll bet it isn't 36 degrees and rainy in Savannah right now! YUK!
First, I'd look at how much voltage that particular converter puts out once the batteries are charged. If it's between 13.2 and 13.6V, I'd just leave it hooked up all the time. That's the typical fixed voltage float charge of most battery chargers I've seen. Just check the water level periodically.
If you disconnect the converter at the AC side, I wouldn't use an exposed knife switch... put a box in the AC line and a switch in it sufficient to handle the current of the inverter.
I wouldn't disconnect the batteries themselves if you're gonna leave the trailer AC live. Many converters have poor, if any, filtering after rectification to DC, and rely on the batteries to absorb most of the AC ripple. Having that going to your 12V
appliances wouldn't be a good thing.