I am trying to come up with a simple way to keep the frozen food frozen in another power outage.
A few years back, I bought a 800W inverter to connect to my car's battery. it can run a 21" TV, the cable modem, the wifi router, a laptop, but it cannot run the refrigerator which draws 11.6 A.
Priorities, eh?

I know exactly what you mean
Here's a useful tip, many of the things you list there will run on 12V, or versions can be obtained that run on 12V. So it's possible to avoid the vampire load of a double conversion.
As far as the refrigerator, you have the right concept. My rule of thumb is refrigerators take 1 kilowatt-hour per day. That's not bad at all, it only averages 42 watts. But it's the startup surge that kicks you in the pants!
I was doing the calculations and for the refrigerator alone, I need a generator that can output 1392 watts but probably more like 2800 watts peak.
Yeah, that's startup current for you! But you're on the right track. You just need to upsize the inverter to handle the startup load.
Or you can take the mountain to Mohammad, and shop for an inverter refrigerator (that is to say: one that has a VFD driving the compressor motor). Those inherently "soft-start", which may be easier to drive from the inverter.
Would it make sense to buy an ice maker and run it on the inverter and fill an Igloo cooler with ice and the frozen food?
No, that would not work at all, it would have the same startup surge problem as the fridge, and the ice wouldn't keep your frozens frozen. Frozens need to be around 10F, your ice will be at 32F. Ice works for the *refrigerator* part, that's all.
Matter of perspective. Smaller than a 22K whole house generator. I don't see the reluctance to go the generator route.
*penguins climb off boat, finally reaching Antarctica*
*they take it all in*
"This sucks."
That's generators for ya. Like any small engine, they are balky and fidgety.
- The battery went flat
- The fuel got old
- The fuel filter is plugged, For, No, Reason
- The carb needs rebuilding.
- The plug is fouled (why??????)
- The magneto quit.
Nothing is "NO" maintenance. Not even my wife

Keep it up and it will run for years.
I knocked together a simple solar system to sustain an old Edison battery (itself 40++ years old). MorningStar SunGuard ($30), 20-watt solar panel, that's pretty much it. Nine years on, and the thing just keeps on ticking. Zero maintenance for something *that* much of a hobby-hack, that's impressive.
Granted, the "emergency generator" application is a very hard one, because you need it so rarely, and when you do it's critical. But this is exactly what small gas engines are particularly bad at.