As others have suggested you need to follow the power. Is the inverter getting 12 volts, is it outputting 120 volts. Is something plugged into it or is it hard wired into the rig. If hardwired can you follow the circuit to see where it connects to the circuit feeding the fridge. You will have to play detective.
We run our Samsung residential fridge from a 600 watt PSW inverter with no problems. It draws 135 watts when running, and if both cooling units, the ice maker is ejecting ice and the defrost cycle is running at the same time it will draw 465 watts. The inverter has surge capacity to 1,000 watts so start up is not an issue.