There has top be a thousand posts on this topic just this year. I'll give you some brief details of what you should look for and then use the archives to read up on which area looks promising.
the system has basically two controllers in series so to speak. The pushbutton controller decides on whether or not to turn the compressor on based upon its inputs.
The compressor on signal should be conducted to the klima relay (compressor control relay) through the low pressure switch. This wiring is real easy to view and is the first step in your diagnosis.
In most cases the signal from PBC to Klima is a ground but sometimes its 12v monitor the signal as you turn the A/C on to verify condition of the signal and whether it goes away when the problem exists.
It will probably stay as your systems indicate to me that the Klima relay is stopping the compressor. It does it to protect the engine and compressor from various mechanical problems. It protects the engine from overheating by shuting down the comp. It protects the belt from a frozen compressor by monitoring the engine and comp speed. In this function comes much of the problem. Loose/slipping belts, slipping clutches due to various oils on the surface, etc etc,
And then the last thing you should look for is the logic and plan to lobotomize the Klima when all else fails.
Bosch Master, ASE Master, L1
33 years MB technician