Due to the inrush on motor starting of the refrigerator compressor it immediately knocked the generator out. How do inverters handle inrush? Will they need to be overrated for motor starting purposes?
Yes, they do. A typical motor will draw 3-4x running current for startup, and some are worse. It depends on your inverter, but most have some limited surge capability, and the good ones have a 3x surge capability... I like my Aims units. They are not cheap, they are not light, idle draw leaves something to desire, but if they can run it, they can start it. And I have beat the hell out of my Aims over the past 6 years of it running 24/7 with only a few brief maintenance windows, including running an awful lot of nasty inductive spot welder...
Most inverter generators have zero surge capacity. What they are rated at, they can start, and there's no overhead to be found. This is why a lot of people with Honda 2000s or similar are getting rid of them cheap - they might run an air conditioner, but the won't start it. You can get a soft start kit for the air conditioner, or a bigger generator, and one involves fiddly wiring, one is just more money.
So, uh... if anyone wants a competent solar power trailer for home backup (with enough inverter capacity to start just about anything), I kinda am starting a company to build them...