To be reliable and long lived systems, including the batteries, need to be designed for the charge and discharge cycles they follow. A typical car battery is designed for frequent relatively shallow discharge cycles, and is probably not a bad match for the way your refrigerator uses power in that respect. But it's going to be the equivalent of starting a car a couple dozen or more times a day (assuming your fridge has to run for a few minutes every hour, at minimum). The battery life will affected accordingly. You would also need to consider the charger = it should be regulated to only charge when the battery is discharged, and to throttle off when the battery is fully charged. Overall, unless you're into designing charging circuits and regulation, just hooking a charger and inverter to an automotive type battery and letting it run is not an optimal solutio Yourn. If you're looking at any other type of battery, same considerations apply, but with different parameters. Your case, e.g., is somewhat similar to what the drive battery in a hybrid car experiences in stop and go traffic. There is a reason for the elaborate electronics and software that manage those batteries.

On the other hand your battery + inverter + transfer solution isn't perfect either, but it's close to the way a lead acid powered UPS would work. As long as you check the battery periodically for ability to sustain a discharge, and top it up as needed, it's a pretty simple solution - not quite fire and forget, but minimally complicated. If it were me, I'd have gone out and bought a real UPS, but short of that, your solutions with the ATS is pretty sound.