OK, I think everything is working fine, and there is some sort of irregularity with the Kill A Watt when using the MSW inverter running the fridge.
I have done two more tests using only the Kwh on the KAW.
8 hours on shore power and the usage was 320 watts. This was my base line.
8 hours on inverter power and the usage was 280 watts. This was overnight, so the fridge doors were never opened.
Based on these readings, I think I can conclude that both the MSW inverter and the refrigerator are working fine together. Since I am expecting about 900-1,000 watts usage per day, I think I am on target, and everything should be fine.
Although I still cannot explain why the Kill A Watt readings for Amps are so different for shore vs inverter power, it does appear over time that the Kwh readings match and are right in line for what I was expecting.
Thanks to all who helped me understand this problem better.
2002 Diplomat pulling:
23' Enclosed Tilt Trailer with:
Car, Boat, Harley, 660 w solar,and 2 bikes