Let $X$ be a variety of dimension $d \geq 2$ (over a field), consisting of two irreducible components meeting transversely in a divisor $D$. (We can assume these components and $D$ are as nice as we like.)
Let $I$ be a rank-1 torsion-free sheaf on $X$ such that $I|_D$ is a rank-2 locally free sheaf on $D$.
Let $Y = \text{Proj} (\text{Sym} (I))$, with $\psi: Y \rightarrow X$ the structure map. (So $E:=\psi^{-1}(D)$ is a $\mathbf{P}^1$-bundle over $D$, and $Y \backslash E$ is isomorphic to $X \backslash D$.)
Let $M$ be an invertible sheaf on $Y$ whose restriction to each fiber of the $\mathbf{P}^1$-bundle $E \rightarrow D$ is trivial. (Sanity check: this assumption implies that $\psi_*M$ is torsion-free and $R^1 \psi_*M = 0$. The same would be true if the restriction of $M$ to each fiber of $E \rightarrow D$ had degree $-1$ or $1$, but in those cases the answer to the question below would be no.)
Question: is $\psi_*M$ invertible?
