You have an interesting problem and whilst I cannot comment on your problem solution as others have directly , I would that what you have is fundamentally unsound in terms of the wiring and should be amended to enhance safety and also reduce power consumption and reduce wear and tear on the fridge unit
I would suggest that this wiring is redone to allow the fridge to work on its own and be its own master. What I mean by this is
1) make the whole of the fridge circuit – thermostat, compressor etc have one source of power (and ground of possible) .
2) this comes directly in two pole format from the switch board
3) if necessary this can come directly from the relay – it is a good idea to let the high current relay take the load rather than the small capability switch.
so in the end you will end up with a switch controlling the relay and providing an isolated and fused supply, and a fridge unit in totality which looks after itself. The compressor should be controlled directly from the thermostatic the toolbox
As for the lack of cooling either the compressor is not working – just the fan running or is it is working then the coolant has escaped and needs degassing if possible, just like in your car Air con…. this may or may not be possible in which case the fridge unit will need replacement.
….. just my thoughts……..