  We show that a finite volume deformation retract
$\mathcal{T}_{\varepsilon_t}^{-}(\mathcal{N}_g)/\mathrm{MCG}(\mathcal{N}_g)$ of
the moduli space $\mathcal{M}(\mathcal{N}_g)$ of non-orientable surfaces
$\mathcal{N}_g$ behaves like the convex core of $\mathcal{M}(\mathcal{N}_g)$,
despite not even being quasi-convex. We then show that geodesics in the convex
core leave compact regions with exponentially low probabilities, showing that
the action of $\mathrm{MCG}(\mathcal{N}_g)$ on
$\mathcal{T}_{\varepsilon_t}^{-}(\mathcal{N}_g)$ is statistically
convex-cocompact. Combined with results of Coulon and Yang, this shows that the
growth rate of orbit points under the mapping class group action is purely
exponential, pseudo-Anosov elements in mapping class groups of non-orientable
surfaces are exponentially generic, and the action of mapping class group on
the limit set in the horofunction boundary is ergodic with respect to the
Patterson-Sullivan measure. A key step of our proof relies on complexity
length, developed by Dowdall and Masur, which is an alternative notion of
distance on Teichm\"uller space that accounts for geodesics that spend a
considerable fraction of their time in the thin part.
