Somewhat belatedly (four years later), I noticed that the Shapley-Folkman lemma (https://en.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma) yields a strong version of the result. SF says that if $S$ is a subset of $n$-dimensional Euclidean space, and we set $P$ to be its convex hull, then when $d > n$, for all $x$ in $dP$, there exists $y $ in $nP$ such that $x - y$ is a sum of $n-d$ elements of $S$. 

 So let $S = P \cap {\bf Z^n}$. For $x$ in $dP \cap {\bf Z^n}$, SF implies there is a decomposition $x = y + w$ where $w$ is a sum of $d-n$ elements of $S$ and $y \in dP$. Then $y = x-w$ is in ${\bf Z^n}$ (since $x$ and $w$ are). Hence $y \in nP \cap {\bf Z^n}$, a stronger result: $dP \cap {\bf Z^n} = nP \cap {\bf Z^n} + ( S + \dots + S)$ with $d-n$ copies of $S$.