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.
Need and want are different many times. With this design, they are closer. I am doing a very simple box. I want it to be livable, with basic needs met, and not be extravagant. Half a dozen lights, one roof vent, one AC unit, 2 sinks (one kitchen one bath), fridge (type to be determined still...I have a couple in storage I wouldnt mind using to start off with), shower and a toilet. A place to sit down to eat or read the paper (dinette shouldnt cost more than a couple hundred in lumber at most..and will double as another sleeping spot in case I have company), the countertop to prepare simple foods, and I am good. I dont watch TV, so no antenna set ups are needed. I have internet off my phone, so that is covered.
I looked at a pure sine wave because of the laptop. I dont know how well the modified sine would interact with the laptop, and I dont want to replace it right away (mostly because I am cheap, lol). And I agree with RanD, the 3k inverter is *more* than enough to run the fridge and some lights. I went bigger because I have bought smaller tools in the past instead of the next size up, and it always bites me in the butt. I looked at an apartment fridge online today (nothing special about it) than ran on 3amps, and the draw was only 90watts. I am sure it's more than 90, as I am sure the one I have in storage is as well. But I am sure it will run fine. In the end, it will come down to what is nice to have, vs is it worth the cost. I dont plan on having hundreds of dollars of food in the fridge. I live alone. So I can shut it down if needed between jumps.
I did plan on using all electric in the house. It did avoid the need for propane. But there is one part of the house that might be better off using propane, and that's the stove. Someone suggested all DC in the house instead of AC, but that didnt make a lot of sense to me, at least not on the surface. Running AC to the truck, converting it to DC to run everything seems odd. Especially when the only thing I have that will only run on DC is the roof vent? Or is the air conditioner a DC item as well? None I have seen so far are DC, but I have been wrong before..