Not sure what the "permissives are" (conditions that must be met before an order will be given) for the AC clutch in your TDI. However, applying the KISS method, start with the basics and work back towards the complex.
With the engine "secured" and the key out of the ignition, disconnect the electrical plug to the AC clutch and take an ohmic reading. You should have a very low ohmic reading. I don't have the TP in front of me, but a coil designed to cause a mechanical movement of this size should be well under 25-ohms. Using R=E/I, and assuming a 10-amp circuit draw, you should see an ohmic value in the neighborhood of 1 to 2 ohms. If you show a very high reading or an open circuit, you simply have an AC clutch that is burnt out and requires renewal.
If there are (2) wires on the harness going down to the compressor clutch, then check to ensure the AC clutch coil is not grounded to the case. Take an ohm reading from the compressor case back to one of the leads, then repeat from the other lead to the compressor case. Both readings should be very high and ideally read an open circuit, greater than 1M ohm.
I am assuming that you have checked all of the associated fuses. Fuses don't always blow or fail in such a way that they are 'visibly' blown. I would check the associated fuses with an ohm meter.
As for the permissives, I would get a circuit diagram in front of you and start walking the circuit. Most permissives will stand out (interrupts the main circuit with decommissioning ability) or there will be a note if they are controlled in the higher order logic of a control module. The (3) most common HVAC permissives are, and they depend on the application, suction-side low pressure (loss of charge), discharge-side high pressure (lack of heat transfer across the condenser, an overcharge, or a clogged TXV or drier...system obstruction), and low oil level/pressure (not very common on automotive applications).
High side obstructions don't cause a full system lock out, unless the circuit is designed to latch. High side pressure shut downs cause short cycling, as the pressure will bleed back down thru the system and then the compressor will make another attempt, but will run for a very short duration.
Automotive applications do include a "wide open throttle" circuit on the AC system, which forces a decommissioning of the compressor clutch to increase the available horsepower at the tires when you put the pedal to the floor. The AC compressor clutch will remain decommissioned until you reduce the throttle.
