I am very interested in this topic although I do not feel qualified to comment. I have no RE so I am interested in saving energy instead. I live in an urban trailer park in the north east USA and the landlord, neighbors, and town are very strict.
That said, it seems the whole matter is one of power factor correction. Super caps across the input to the inverter are a great idea, but what capacity? You can order about a farad at 10 - 16 volts from digi-key or mouser.com for about 60 USD and do some wiring in series/parallel like batteries. And they would be good for about 5 time constants till they reach 0.63 charge, but is that enough for 40 seconds? My brain is fried too much for the math at the moment, but I'm very interested.
The other way to do it is inductance. Both are a form of energy storage, and an inductance will oppose a change in current, which seems to be the desired outcome. Hence get a solid bar of steel and wrap it with enough wire and place it into one of the "hot" legs... put it on the inverter output instead. But how much energy to store at (assuming) the bog-standard 60 cycles? And for what length of time constants?
Now I gotta get pencil and paper and think.
You are trying to PF a fridge which was a brilliant idea, by the way to convert a chest freezer. And that is a lot easier to do with sine waves and a known inductance for the motor. It's a bunch of maths, and the power company will hate you.
That said, let me go and chew on this a bit. Good luck! Also remember: caps and coils will eith lead or lag the voltage or current draw, depending. At a given frequency. Very likely you'll need to measure and do the maths for each individual appliance to make it work, but its definitely a good possibility. Old technology but nobody else bothers anymore. I would love to "power factor" (PF) my entire house but that is a project for later.