3000W is not only big enough to run the fridge, it is big enough to run the air conditioner with room to spare. 3000W is 25A at 120VAC. A Coleman polar cub draws about 11A to 13A depending on how hot it is. Such a big inverter is expensive, which is why you don't want one like that. You probably need something closer to 3A for a refrigerator. That's only 360W. Don't take my word for it though, check the specifications, or better yet measure it yourself. Make sure you measure it in all modes of operation, including when the compressor first comes on. Most inverters are rated to handle more than the nominal current for a short time. Yes, a modified sine wave will probably run a refrigerator. Many have done it and perhaps some of them will chime in. I just can't make any guarantees. Motors run hotter and less efficiently if not fed with a sine wave.
Yes propane fridges are fine, and many use them. Thus far you have avoided the need for propane and all that entails. All electric (no propane) is getting more and more popular though.
All these systems and design decisions interact. More AC requirements, bigger inverter, bigger batteries, more cost. At some point you need to nail down the line between what you need and what you want. Once you define your mission parameters the rest of the tradeoffs can be calculated. Then you convert mission parameters to a design, which converts to cost, and forces you to rethink your mission.
I'ts entirely your decision of course, but I might start with batteries and inverter that only support your refrigerator and a few minimal lights for 24 hours maximum, and that's only drawing down the battery 50%. That should get you from point A to point B where you can plug in and charge up, or at least run the generator for a while. Once you have a rough idea of the cost you can go from there.
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