The narrator is at great pains to stress that the story happened a long time in the past. Elsewhere in the Foreword, he says:
Diese Geschichte ist sehr lange her, sie ist sozusagen schon ganz mit
historischem Edelrost überzogen und unbedingt in der Zeitform der
tiefsten Vergangenheit vorzutragen.
This story happened a very long time ago. It is so-to-speak already
covered in historical rust and must be related in the tense form of
the deepest past. (my translation)
The German passage, of which you quote the translation, is:
Sie spielt, oder, um jedes Präsens geflissentlich zu vermeiden, sie
spielte und hat gespielt vormals, ehedem, in den alten Tagen, der Welt
vor dem großen Kriege.
My translation of this is:
It (the story) takes place, or in order to deliberately avoid every
present tense, it took place and has taken place back then, long ago,
in the old days, the world before the Great War.
So the intention of the narrator/author is very clear here. The translator has turned the literal "in order to deliberately avoid every present tense" into "to avoid any present tense whatever". This accurately conveys the tenor of the German by using the intensifier whatever.
Intensifiers include common words such as very, really, awfully which precede the words they modify. They also include words or phrases which follow the words they modify or qualify such as whatsoever, at all, like anything.
The narrator plans to use no present tense whatsoever or at all.
Here is an article discussing the intensifier whatsoever.