So, basically you want to modify a fridge with an electromotor-driven compressor in it to use propane instead of electricity to drive it.

This is not exactly a trivial task. You could switch to absorption cooling (the intermittent cycle using NH3 is easy), or use an automotive compressor coupled to a small 4-stroke fuel engine running on propane.

Another option is to inject liquid propane into the evaporator and dump the evaporated propane either in the atmosphere or route the evaporated propane to other propane equipment. However, this causes a serious consumption of propane since you are only using the latent heat of vaporisation instead of the chemical energy.

I would go either for the fuel engine (small lawnmower?) with external compressor, or the intermittent NH3 cycle