When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?

This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.

Let $X$ be a regular 2-dimensional Noetherian scheme, for example an arithmetic surface $\pi:X \to S$.

Suppose we have line bundles $\mathcal{L}_1, \mathcal{L}_2$ and a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ satisying$$\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2.$$

Question: When can we conclude that $\mathcal{F}$ is locally free?

As submodules of a locally free sheaf on a smooth curve are locally free, this would hold if $X$ were a smooth curve. We do know at least that it is torsion-free.

Really the most important thing I need to know is when $\mathcal{F}$ will preserve exactness of sequences$$0 \to \mathcal{L}' \to \mathcal{L} \to \overline{\mathcal{L}} \to 0$$$$0 \dashrightarrow \mathcal{L}'\otimes\mathcal{F} \to \mathcal{L}\otimes\mathcal{F} \to \overline{\mathcal{L}}\otimes\mathcal{F} \to 0$$

for line bundles $\mathcal{L}', \mathcal{L}$.

So flatness would suffice. But by this and thisI believe flat is equivalent to projective in this situation.

In lieu of flatness, a characterization of the $\mathcal{L}$ for which $\mathcal{T}or_1(\mathcal{L}, \mathcal{F})$ vanishes could potentially satisfy my needs, but knowing $\mathcal{F}$ is locally free would be preferable.

Any help would be appreciated.