The Fall of Babylon in 539 is attested biblically because the Bible not only foretells that event but also desribes the event in detail. This is the only pivotal date that is descriptive of an event which took place for other pivotal dates are not mentioned in the Bible but are simply based on secular records for a certain year for a certain king etc.
As usual, 'scholar's' reasoning is flawed.
The fact is that because the Society does not give credence to astronomical diaries, the Society - according to its own reasoning regarding secular authorities - has no basis to accept 539 for the overthrow of Babylon. Regardless of how specifically the bible describes the event, it simply provides no clue as to an absolute year. The Society indeed relies on the so-called 'profane' chronology based on astronomical diaries to get 539, but then proceeds to ignore the professionals for everything else, in preference of their outlandish reworking of history.