  We prove a converse theorem for the multiplicative Borcherds lift for
lattices of square-free level whose associated discriminant group is
anisotropic. This can be seen as generalization of Bruinier's results in
\cite{Br2}, which provides a converse theorem for lattices of prime level. The
surjectivity of the Borcherds lift in our case follows from the injectivity of
the Kudla-Millson theta lift. We generalize the corresponding results in
\cite{BF1} to the aforementioned lattices and thereby in particular to lattices
which are not unimodular and not of type $(p,2)$. Along the way, we compute the
contribution of both, the non-Archimedean and Archimedean places of the
$L^2$-norm of the Kudla-Millson theta lift. As an application we refine a
theorem of Scheithauer on the non-existence of reflective automorphic products.
