12V PWM controller for 12V fan that gets faster with the COLD
You cannot simply invert the output of a controller. Then the thermal negative feedback loop becomes a positive feedback loop and latches in one direction.

The inputs must be reversed with thermal negative feedback so that the thermostat regulates the cold temperature or minimum value rather than the hot.

====================================================

You need to avoid adding heat with a fan on the cold side yet remove as much heat as possible on the hot side. 

THis is a simple way to make a cold thermostat.







simulate this circuit &ndash; Schematic created using CircuitLab

A better approach may be to maximize thermal conduction on hot side with continuous forced air exhaust to remove heat and cold side with large  passive heatsink and then regulate Peltier device current for regulating cold side temp.  Efficiency improves by reducing the temperature drop.