You do not quite understand the significance of the readings at the low pressure switch.
According to what you have said it would appear that your problem is the pushbutton controller.
The key here is what is taking place at the low pressure switch. The switch is the conduit for the "ON" signal from the PBC (pushbutton controller). That "ON" signal is a ground signal.
One needs to measure the voltage with the system hooked up. The Klima sends out a 12v signal and the PBC grounds it. If one read 12v on one side and nothing on the other the switch would be open. If one jumped it the resulting point will either look like 12v (if the PBC is NOT grounding) or 0v (if it is grounding - actually it won't pull it all the way to 0v). Anyway to remove the PBC from the decision, ground the Klima signa at the low pressure switch.
One must watch out using these instructions as the similar systems of some diesels are different and they send a 12v signal from the PBC to the Klima instead. Grounding that circuit will burn a circuit board in the PBC.
Bosch Master, ASE Master, L1
33 years MB technician