It´s a start...
Normally the switch must be closed when there´s enough cooling gas in the system and the switch works. The switch brings battery voltage to the relay for the clutch. You wrote that the A/C control unit wants the compressor to work. Can you measure battery voltage on one of the pins of the switch? (ignition on)
What´s about the thermostat near the air intake? The relay doesn´t work when it´s switched of.
Excuse my very bad english

Gruss, Georg Hubraum statt Wohnraum..