LG fridge is on the fritz

It looks like the compressor in our LG fridge has given up the ghost. The fan works, the coils are clean (I checked and cleaned them again just to be sure). I unplugged the fridge for more than 12 hours; when I started it again I could hear the motor clicking every few minutes, so it’s trying to start - just not able to.

I’ve spoken to a couple of repairmen and they’re pretty sure it’s my compressor, although I’ve been reading up on it and there’s also a start relay. Could that burn out and cause the problem?

LG refuses to sell parts to anyone but LG technicians so this means that my local guy is not allowed to touch my fridge. I live on a farm and it’s at least an hour’s drive for the approved technician to get here. What kind of cost am I looking at by the time he gets here, replaces the part etc.?