Do you like math? I Do.
Look at the label on the fridge. I will use a small Dorm/Office type that sucks 100 watts running, about 500 starting.
Now you said a SINGLE 12 volt battery. This means a MARINE/rv or a MARINE/deep cycle. you can only use about 25% of the power in the battery
Group size: Amp hours. Usable watt hours (At the 20 hour rate)
24 - - -- - - 75 - - - - - - 187 so it might run about an hour.
(NOTE I use 10 volts as the battery voltage in these calculations to allow for inverter loss)
27 - - - - - 95 - - - - - - -237.5
29 - - - - - 105 - - - - - 262.5
31 - - - - - 130 - - - - - 325
4d - - - - - 230 -- - - - - 575
Now if you go to Golf Car Batteries (NOTE the numbers here) GC-2
YOu can get GC2 in the 230 amp hour size as well but you can use fully HALF of that power so now you have 1150 Watt Hours for that fridge.
divide usable watts by fridge watts to find the result.... Figure aboutr 3/4 run time to offset the surge.
Note that there is an effect. Can't recall the name (Pulket???) exactly but what it says is the faster you suck out the power the faster still the battery runs down So if you draw at the 20 hour rate you will get performance close to what I listed. but draw at the 10 hour rate and your run time goes down FASTER (You get like 1/4 the run time)
Bottom line.. Get a generator.