  A general curve $C$ of genus six is canonically embedded into the smooth del
Pezzo surface $\Sigma \subseteq \mathbb{P}^1 \times \mathbb{P}^2$ of degree $5$
as a divisor in the class $\mathcal{O}_{\Sigma}(2,2)$. In this article, we
study the variation of geometric invariant theory (VGIT) for such pairs
$(\Sigma,C)$, and relate the VGIT moduli spaces to the K-moduli of pairs
$(\Sigma,C)$ and the Hassett-Keel program for moduli of genus six curves. We
prove that the K-moduli spaces ${\overline{M}}^{K}(c)$ give the final several
steps in the Hassett-Keel program for ${\overline{M}}_6$.
