Have you seen my matroid?

One can use Whitney's theorem to show that the characteristic polynomial is $$ \sum_{i=0}^k{n\choose i}q^{k-i}(q-2)^{n-i} + \sum_{i=k+1}^n{n\choose i}(q-2)^{n-i}. $$ I doubt that this can be simplified.

Let $U$ be the uniform matroid of rank $k$ on $n$. Since $U$ is orientable one can consider the Lawrence oriented matroid $\Lambda(U)$ associated with any orientation of $U$ (the Lawrence construction doesn't care about which orientation you take). Then $M(n,k)$ is precisely the underlying unoriented matroid $\underline{\Lambda(U)}$ of $\Lambda(U)$.

Also, the dual matroid $M^*(n,k)$ is a symplectic matroid, which explains why the group $B_n$ acts on the primal.