The very last battery brand I would buy any more since being purchased by Johnson Controls is Optima. I had three (3!!) Optima batteries fail prematurely in a row... two Yellow Tops and one Blue Top. Johnson Controls moved Optima's entire manufacturing operation out of the US and into Mexico which is when the problems started. Several Optima distributors tasked with collecting bad Optima batteries replaced under warranty said the problem is caused by having gone with lower grade components like a substantially less pure lead.
But to your question, how long a 12v battery will last powering a refrigerator depends on the refrigerator. High-end refrigerators like made by ARB can run several days on a battery. Cheap refrigerators like I tried in my trailer can drain the battery in less than a day.
Plugging the battery into the cigarette lighter is not going to keep the battery charged, it can draw way too much power which will pop the fuse. Plugging most refrigerators into the cigarette lighter will work but be wary which one you buy. As said, some draw a lot of current, better more expensive models don't draw much at all.
About the only type of refrigerator I'd even consider for keeping plugged in for several days is an ARB or one of its close cousins but they're really expensive. https://www.google.com/search?hl=en&...32.vinbk1hyyRA
If that ARB class refrigerator is too much $$$, which it is for me, consider a good cooler like the Coleman Igloo. My Coleman Igloo keeps everything REALLY cold for my 3 day desert trips with a big block of ice as its central component. It even managed to keep that block of ice in good enough shape that I would put it back into my freezer for the next trip. My last block of ice lasted 3-4 years before I retired it.
P.S. 12v refrigerators like the ARB are so good they can keep ice cream rock hard.