Projections to the second summands define a morphism from that complex to the complex
$$
F \stackrel{s}\to F(D)\tag{*}
$$
of locally free sheaves. The cone of this morphism is the complex
$$
0 \to E \stackrel{s}\to E(D) \stackrel{\rho_E}\to E(D)\vert_D \to 0
$$
which is acyclic. Therefore, $(*)$ is a locally free resolution of the original complex.