For one thing, the a/c compressor clutch shouldn't engage if the internal fan is set to '0' (ie off). This is to prevent frosting of the evaporator coil and the system cycling on it's low pressure switch (or frost sensor if it has one). It is also pointless running an a/c system with no air to cool or dehumidify.
These cars run on R134a refrigerant, which is actually a 'medium temperature' refrigerant. It has been widely used (until very recently) in refrigeration equipment which maintain approx 0-4c temperatures. So, in reality, your cars don't have a/c; they have 'fridges!
I will admit I am not 'au fait' with the nuances of the Z3, but the above should be the case on any car with a/c.
The only 'noise' you should hear from the A/C system is a 'click' as the clutch engages and occasionally a faint 'rushing' type of noise at the expansion valve (near the bulkhead somewhere which can sometimes be heard inside the cabin). This can become slightly louder if the system is low of refrigerant, but has enough to continue to operate. Not loud enough to be heard over a running engine with the bonnet open, though. It would be constant, not just a quick 'hiss' every so often.
The noise in the video (my PC speakers are not the best admittedly) is far too loud to be anything a/c related, unless there is a problem with either the drive belt slipping momentarily when the clutch is engaged, or the compressor itself has a major problem.
Of course without actually seeing what it is doing, the above can only be an educated guess.