An inverter will not let you run your fridge until the battery is "dead". It's going to have a low voltage cut off, likely somewhere around 11 Volts, specifically to avoid damaging batteries by fully discharging them.
How many hours you'll get from the battery mostly depends on your ambient air temperature and how often you open the fridge. They don't use that much power when they're idle - my fridge averages at about 90 watts (I'm not running off grid, but I do have rooftop solar and our system produces pretty charts showing consumption). A large car battery can sustain 90 watts for a quite long time - well over 2 hours. Probably closer to 10.
Running a fridge off a car battery long term is a bad idea. But in an emergency? Sure I'd totally do that - especially if your "emergency" is genuine such as needing to keep your medication cold. Just don't open the fridge unless you're taking your medication.
LifePo4 FTW!
Sure. Way better than lead acid. But that doesn't mean lead acid is useless. When I lived off grid, LifePo4 didn't exist and we got close ten years (of daily use) out of our lead acid batteries. They were bigger than car batteries and also deep cycle ones, but in an emergency a car battery would be a fine choice if it's the best one you have.