There is only 12V at C4 when the compressor relay closes, which is controlled by the ECU. Try jumping the relay contact by bridging brown/white wire to the red at the AC control module to see if the compressor starts.
If it does, it could be the Trinary switch not signalling the ECU to start the compressor due to low refrigerant level or defective wiring or switch.
The fans are also controlled by the ECU and run at half speed if the compressor is running.
I bridged out the C4 fuse and I could hear the A/C compressor solenoid kick in.
Could this still mean that it is low on gas in the A/C?
I know the switch is ok but the back lights don't work on the heater control panel either. Could this all be linked to the control box under the front wing?