The pushbutton controller grounds the Klima relay through the low pressure switch. Verify that this is happening at the low pressure switch.
If the signal is proper, then the various signals should be checked at the Klima relay. I don't know how the book says to do it, but I bypass the Klima relay (with it out) and that makes the compressor turn. With it turning one can then verify the compressor speed signal. The engine speed signal can be verified without the comp jumped.
Bosch Master, ASE Master, L1
33 years MB technician