To answer your question about a by-pass.
There are a number of threads giving actual pin numbers but here is the concept. The Klima is a smart relay that does what all relays do. It connects the clutch coil to battery voltage. Look to the bottom of the device and the power in will be term 30, power out is likely to be term 87. Verify that powering term 87 runs the compressor.
Your dumb relay will have to connect these two wires. The Klima receives its command signal from the PBC (push button controller) as a ground signal. So a properly wired 4 pin relay will have power on 30, output on 87, ignition power from a term 15 placed on term 86 and the command signal turning the relay on with a ground signal placed to the dumb relay term 85.
Before you waste a bunch of time verify the ground signal at the low pressure switch (the compressor on ground signal passes through the low pressure switch on the drier on the way to the Klima from the PBC).
Bosch Master, ASE Master, L1
33 years MB technician