I am not certain, but I suspect the the original thermostat was in the small freezer at the top of the mini fridge, and now the unit is trying to cool the whole mini fridge to that lower temperature with the cooling plate.
So if you plug in an external temperature controller like a johnson digital controller and control the fridge with that you will need to make sure the original thermostat is set to a lower temperature (cooler) than the temp that you are trying to maintain with the aftermarket temp controller. turning it to max cold would be an effective way to make sure the contacts would not interrupt the circuit.
if you are using an ebay temp controller you will need to wire the contoller to constant power and common to power up the device, and then run a jumper wire from the 120v hot wire to one of the contact terminals on the controllers internal relay. the other side of the relay contact will be wired either to the switch on the existing thermostat, or the compressor hot lead if you choose to leave the existing t-stat.