The inverter is usually designed to cut out if the voltage in inadequate to produce proper 120v, 60 Hz, power. A typical cut-off point is 10.5 or 11.0v. If you have only one battery, it doesn't take much load (amps) to pull the battery down. That's because it takes about 11 amps @ 12v to produce 1 amp @120v.
Are you sure your outlets are even wired to the inverter? An inverter is not standard in that trailer - it is normally equipped only with a converter/charger, which produces 12v power for house use and battery charging while shore power is active. An inverter would be an aftermarket add-on, so the outlets powered, if any, would depend on how the installation was done.