  We describe a compactification by KSBA stable pairs of the five-dimensional
moduli space of K3 surfaces with purely non-symplectic automorphism of order
four and $U(2)\oplus D_4^{\oplus2}$ lattice polarization. These K3 surfaces can
be realized as the minimal resolution of the double cover of
$\mathbb{P}^1\times\mathbb{P}^1$ branched along a specific $(4,4)$ curve. We
show that, up to a finite group action, this stable pair compactification is
isomorphic to Kirwan's partial desingularization of the GIT quotient
$(\mathbb{P}^1)^8//\mathrm{SL}_2$ with the symmetric linearization.
