  In this paper we homologically construct a (functorial) BGG resolution of the
finite-dimensional simple module of the nilBrauer algebra by using
infinity-categorical methods following the reconstruction-from-stratification
philosophy, e.g. appearing in Ayala-Mazel-Gee-Rozenblyum. To do so, we prove a
fact of independent interest, that half of the nilBrauer algebra is Koszul.
This BGG resolution categorifies a character formula of Brundan-Wang-Webster.
More generally, we have a (functorial) ``BGG spectral sequence'' which
converges to any desired module; this spectral sequence is secretly a
resolution when the desired module is finite-dimensional. This spectral
sequence also categorifies the character formulae of Brundan-Wang-Webster for
any (possibly infinite-dimensional) simple module. We expect the methods used
here for producing BGG resolutions to be applicable to other (graded)
triangular-based algebras also, especially diagrammatic ones.
