I'm wondering if anyone has ever done this. If the range of the internal temp control could simple be modified a bit that seems like it would be simpler. I don't know where the electronics are located and how hard they would be to get at.
Most fridges have a very simple electro-mechanical thermometer, probably a bi-metallic strip arrangement. The temperature range is dictated by the types of metals and their physical size & arrangement. The mechanism should be very close to (and likely attached to) the knob used to set the temp.
These systems aren't very precise to start with, and are probably difficult to get working reliably outside of their design range.
I looked into this with the thermostat (a simple bimetal deal like BLL said) on my Danby. I took it apart and did find what I believed to be a couple adjustment screws. In the end, I couldn't find any info on how to adjust them or what they did and then I found a PID controller for super cheap on ebay anyway.
If you have any luck with this, report back, though!