Zero Breeze says their unit is rated at 2,300 BTUs. This compares to about 11,000 to 15,000 BTU for most RV air conditioners. So, the Zero Breeze has less than 1/4 the cooling capacity of a standard RV AC unit.
It draws 10 amps at 24 volts. That would require some specific wiring in order to operate from the trailer battery. Converting 12 volt power from the trailer battery to 24 volt to operate the Zero Breeze would require a step-up transformer.
In broad terms you might get about two and a half hours of operation from a single conventional lead-acid 100-amp-hour 12-volt battery, probably somewhat less.
Or, it could apparently be run off of their shore-power adaptor. The transformer brick on their power supply has a limited cord length to the air conditioner to handle the high 24-volt current, so the brick would likely end up inside the space that you are trying to cool.
Either their power supply or a step-up transformer will produce heat, further limiting the cooling capacity.
It is low capacity unit, not inexpensive, and operating it would require making suitable provisions to power it and to provide ventilation.
I strongly suspect that purchasing this unit would not be a very satisfactory experience as the cooling capacity and duration would be very limited.
The reviews on their web site describe some additional limitations users have encountered which you should also consider.
Tip: Due to the Mark 2 portable and low power consumption features, with 2300 BTU cooling capacity, The Mark 2 can cool down a single target in the open outdoors or overall cool down the bedroom area of an RV or truck at night for a comfortable sleep. It is not recommended to replace the family room air conditioning for the overall space cooling or applied to the car under the sun, cabin, and other space cooling.