Like a mountain
The structure, being 4km high and shaped like a pyramid, would act in much the same way as a mountain.
We have plenty of data to draw from, with numerous mountains of similar height and shape to study. Keep in mind Everest is 8000m above sea level, our new building is half that height.
Common weather patterns include:
Precipitation to one ...
Your flying insect-like creature sounds like a bee or wasp. Honeybees chew wax secretions from their bodies to attach the wax flakes to the comb, gradually building a hexagonal lattice comb. Wasps do a similar thing with wood fibers and their own saliva. This would suggest that bee-nest-like structures would be a likely design.
This makes sense ...