Study the diagram in Post 49. Pay attention to where the wiring goes beyond the pressure switch. The diagram shows wire colors and connector pin numbers. You need to follow it all the way back to the Pushbutton Unit. You should be able to test the whole system by grounding Pin 6 (blue/green) on the X1 connector on the PBU. If the clutch engages, the PBU is the problem. If it doesn't, somewhere between the pressure switch and the PBU you have an open connection.

I have no idea what the "F1 Electrical Center" is, but if it's anything like other electronic "modules" of the era, I'd be willing to be you have broken/cracked solder joints.

The PBU grounds Pin 10 of the Klima relay through the pressure switch to engage the compressor clutch. Since you can ground the circuit manually at the pressure switch, you've ruled out the Klima relay and the wiring up to the pressure switch. Now you need to prove the circuit the rest of the way back to the PBU.

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Current stable:
1995 E320 149K (Nancy)
1983 500SL 120K (SLoL)

Black Sheep:
1985 524TD 167K (TotalDumpster™)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)