I put a timer power switch on my old mini beer fridge that had its thermostat break, and just adjusted the time until the internal temp was what i wanted without freezing my beers.
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I used a temperature controller to regulate temps for a freezer I turned into a kegerator. It was a Johnson Controls model, iirc. Not sure what similar options would be outside the US though, if that’s a thing for you.
Maybe check out a local home brewing shop? Looks like on average, prices range anywhere from $50-$500US depending on how fancy you want to get.
You could even use the 'smart' temp sensor and plug to power it on when the temp rises above the desired value, but probably won't be buyable in your local hardware store... Also, R-134a and other refridgerants are vety potent greenhuse gases, so maybe don't release that...
You should be able to find a universal thermostat at a local hardware store. You might even salvage a working thermostat from a fridge with a broken compressor. That would be drastically easier than moving the compressor and messing with the coolant lines. A timer would work, but if it gets too cold you will have problems with icing on the evaporator.