Idling a car for a long time would need similar precautions to running gas generator. It should be as far as reasonably possible from an enclosed living space. Consider Carbon Monixide alarms to be extra safe. As for how to keep the car from being swiped that is going to depend on your situation, If it was me I would park it in the back yard behind a locked gate.
As for inverters Sinewave ones are a little more pricey (but not as bad as they were a few years ago). They are better at running motors and sensitive electronics. It really depends on what you are wanting to run as to the size. But I will say the start up surge on a fridge compressor can brutal. A few years ago I had the house running on a 48V Exeltech 1100W inverter and sometimes it would fail to start the fridge which only drew 175W when running, even when there was very minimal additional loading on the inverter.
When we replaced that fridge we got a Samsung inverted drive one which has very soft starts and can easily run on a 300W DC/AC inverter.
I can't vouch for this
1500W Sinewave inverter, having never seen it, but the feature set and price point look right, Sinewave, information display, remote information display, USB-C PD charge ports, Wireless on off remote, battery cables included.
It would be an interesting experiment to gas up the car, park it in the back yard and try running your needs off it for day or two, cycling it on and off to find the duty cycle that supported your needs. Perhaps start with something like 1 hour on, 2 hours off. The Inverter should have a low voltage alarm and then a low voltage shutoff to protect the battery. When the alarm goes off shut down the inverter because it will be unhappy running things with car starter turning over. Start the car for an hour or so and repeat. At the end of your testing go gas the car back up and determine what your actual burn was.
As for small DC compressor fridges Dometic was the original manufacture of that concept, now dozens of companies make them at lower prices and quality. Probably going to have to rely on your Amazon review BS detector.
The car and ~$200 inverter is probably your cheapest option, but if it was me I would probably be looking at a 12V fridge and a 500-1500Wh power pack with 200W+ of solar. If you are handy you can put your own together for cheaper but it will require more hands on configuration and management. The big power packs already have MPPT Solar charge controllers in them, and make for a much simpler user experience.