Note that curvature in sectional directions tangent to the surface, say $\Sigma$ vanish. It seems sufficient to conclude that the usual Peterson–Codazzi equations hold for the surface. It follows that there is a convex surface $\Sigma'$ in the Euclidian space that is isometric to $\Sigma$.
Now we want to use bow lemma (diffgeometric analog of arm lemma). The bow lemma has a version for Hadamard spaces. Applying it to plane arcs in $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-noncontracting.
Applying Reshetnyak majorization theorem, to plane sections of $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-preserving.
Finally, applying Kirszbraun, we get that the map $\Sigma'\to\Sigma$ extends to a distance-preserving map from the convex hull of $\Sigma'$.