The clue to your question regarding the conjugation choice of 「かかれ」 over 「かかる」 is already in the quote. The key word here is "parallelism".
The speaker makes a request as to how Momose-san should go/come to the event. The speaker first says 「軽い気持ちで来てください」("Just come over casually."), which is clearly in a request/imperative form. Then, s/he adds to say 「決してナメてかかれというわけではなく・・」 ("I don't mean to say 'Look down on it/them!' or anything but..."), the first part of which （ナメてかかれ） is also in an imperative form.
That is parallelism, where the second (and third, etc.) words/phrases naturally follow the footsteps of the first as far as grammar.
That is two requests, or rather one request expressed in two different ways -- with the first one decribed more generally and the second , more specifically. Please note that in this case, the second part is not a real request as it says "I won't say (second request)" but the author still maintained the grammar form parallel to that of the first request.
What would happen if one chose to say 「ナメてかかるというわけではなく」 instead? Nothing dramatic, really. It just would not flow as smoothly as with 「かかれ」 because it would lack parallelism. The native ears in any language just do not miss those little things.