I do not know what is your problem but I can give you some pointers. Check out post #9 in the following post if the compressor stops after a 'fix' period of time. I fixed mine.
W124 - stumped by the Climate Control Unit
This is what I recommend you to do:
1) Set A/C on ( do not use defrost ) and check whether the compressor engages or not.
2) Check whether compressor times out and quits. If it does then measure the time and start the car again and repeat.
3) If it times out like clock work then it is the thermostat inside the evap box.
I think the A/C system principle is all the same for W124, W126 and W201. Give it a try and it costs nothing.
4) You can also try to by-pass the Kilma. I can make you a by-pass Kilma which by-passes everything except the freon low pressure switch. This will give you an idea whether the cut out is caused by belt slipping or engine too hot.
Good luck.