You really need the FSM wiring diagram to understand the system, but I'll do my best, leaving out much of the description since the system IS turning on the compressor to start with.

The Climate Control Unit (CCU) monitors about a half dozen temperature sensors and sends a signal THROUGH the low pressure switch to the Klima (Deutsche for climate) relay behind the battery. The Klima relay monitors the speed sensor and the signal that comes through the low pressure switch and decides when to energize the contacts which engage the compressor clutch.

The monitoring of the RPM sensor is done as magic inside the Klima relay. One can bypass the speed sensoring function by using a standard ice cube relay in place of the Klima relay, but it takes some innovative electric fabrication work due to it's location. I have never done it, but you could gut the Klima relay and use it as a mount for the ice cube relay. That way you could, on the bench, do the soldering in place of the relay or a relay socket onto the Klima relay base, then plug it into its original location.

Before doing all this as a shot in the dark, make sure that you are getting a signal from the low pressure switch to the Klima relay and still have that signal after the clutch has kicked out. This is where you need the FSM wiring diagram to know the pinout for such testing, and of course how to wire up the ice cube.

Hope this mess helps.