[RMcNeal]

If you have 120 VAC at the outlet from the inverter, then the fridge should be running. Does the light inside come on? The inverter should show you the voltage supplied and the amps being used. When mine is good and cold, it cycles very little and uses very little juice. Unless I forget and leave the ice maker on.
I guess it could be that the transfer switch is not working. It should be the next thing in line that the plug from your inverter is going to. Make sure you get 120 VAC out of that.