Ok if you have linked out the thermostat and the compressor is only running for about 30 seconds or so and stopping again then its going to be the compressor at fault.
As by linking out the stat this sends a direct live feed to the compressor which should keep the compressor running until it gets to hot and cut out which is usually after the compressor has been running for a few days none stop
So from what you say it will need a new compressor. There is a relay on the side of the compressor which can fail but if its running for about 30 seconds or so then it wont be that so its the compressor here at fault.
Sorry its not good news but if you need anymore info just get back to me.
I thought as much. Is it difficult to pull out the compressor on these integrated fridges
Also am I only wasting money fitting a new compressor on a 13 year old unit compared to the cost of a replacement fridge/freezer
Im afraid a compressor replacement isn't a diy fix as when you cut the old one off you need to lokring the new pot on and then regas it back up.
But to be honest at 13 years old I wouldn't even mess around with it as a new compressor you are looking at about the £200 mark maybe a little more.