I took a look on the Frigidaire website for the schematic for your fridge, but it looks like the model number you gave may be incorrect.
Essentially, what you want to do is to follow the wires of the temperature sensor back into the guts of the fridge. Somewhere, they will encounter the thermostat. You want to unplug the thermostat and have your temperature controller directly control power to your compressor. If you're lucky, there is already a relay in place to do this and you won't have to touch 120V.
By the way, if size is your main concern, Auber does make a 1/32 DIN unit that's pretty small. It might suit your needs.