I would check the compressor pressures with a set of gauges, verify that the pressure is at saturation pressure for the ambient temperature. The pressure must be over about 70 psi to reset a low pressure trip at 15 psi.
The fact that your FCM is able to do one of its jobs - run a high speed fan - argues for its good health, but is not conclusive.
The fact that your FCM can shift one fan to fast speed when requested, but is not doing so all the time, indicates that the signal for AC off and fast speed fans from the ECU on T14/13 is not present.
The ECU could still be grounding T14/3 to prevent AC operation. This is not a frequent issue, but has been found.
The G65 could be reporting pressure too low, even if your pressure check shows it is actually high. If the G65 pressure sensor is oily on top near the electrical connections, replace it. DO NOT try to jumper the pressure sensor. Even though VW calls it a switch it is not.
One potential issue, seen fairly often, is that the fans drag the voltage down too low. Low voltage supplied on T4 will interlock the system off. You have proven that it is ok with the fans not running (T4a unplugged), but you don't know what it is with the fans powered on, potentially dragging the voltage down. I would unplug the fans, turn on the engine and AC, and see if the compressor engaged. If it does, do not run it long or it will overheat and overpressure your AC system. if the compressor engages during this test, look for what is dragging your voltage down - usually a high resistance connection.