Poisson approximation of random lattices

Fix a subset $S\subset {\mathbb{R}}^n$ of volume at most $cn$ that satisfies $S\cap (-S)=\mathrm{\varnothing }$. We consider two point processes in $S$: the first is the Poisson point process of intensity one, and the second is the restriction of a random lattice to $S$, where the random lattice is distributed uniformly in the space of covolume-one lattices. We show that the total variation distance between these two point processes is at most $C{e}^{-c^{\mathrm{\prime }}n}$, where $c,C,c^{\mathrm{\prime }}>0$ are universal constants.