We study a dilute gas of interacting fermions at temperature $T=0$ and
chemical potential $\mu \in \mathbb{R}$. The particles are trapped by an
external potential, and they interact via a microscopic attractive two-body
potential with a two-body bound state. We prove the emergence of the
macroscopic Gross-Pitaevskii theory as first-order contribution to the BCS
energy functional in the regime of vanishing micro-to-macro scale parameter.
