Most inverters are only 80% to 85% efficient and many aren't rated even that high.
Have you determined how many amps you are actually drawing from the battery when running the fridge on the inverter?
If you figure 85% efficiency the 2.4 amps when on shore power looks like
1.22 x 2.4 x 120 volts = 352 watts. So on the 12 volt
side it is 352 / 12 = 29.3 amp draw on the battery.
If for example you were trying to run a 240 watt heating element with a 12 volt
battery, you would require a continuous 20 amp draw at 100% efficiency. Running the fridge on 12 volt
electric (battery) is really inefficient.