I have two generators that I use for my work One is a 2200 watt and one is a 5000/6500 watt. I plugged the fridge up to the 2200 watt generator thinking it would power it and it barely did it, it did not shut the generator off but you could tell it was almost under full load while running.
My 5000 watt one was about $599 from home depot and the 2200 watt one was $329 and from Northern Tool and that includes a 5 year no questions asked warranty. They said if it breaks bring it back and get a new one. I have had the 2200 for about 10 months and the 5000 for 2 years both have been good and I use them almost everyday.
I used to work for a guy who had a GMC 2500 and had a 3000 watt inverter wired directly to the battery. I used it a few times to run a skil saw and the saw would bind up because it did not produce enough power.
I would not use an inverter for these purposes.