  We describe the 6-dimensional compact K-moduli space of Fano threefolds in
deformation family No 2.18. These Fano threefolds are double covers of $\mathbb
P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and
Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this
family is K-stable. A member of family No 2.18 admits the structures of a conic
bundle and a quadric surface bundle. We prove that K-polystable limits of these
Fano threefolds admit conic bundle structures, but not necessarily del Pezzo
fibration structures. We study this K-moduli space via the moduli space of log
Fano pairs $(\mathbb P^1\times\mathbb P^2, c R)$ for $c=1/2$ and $R$ a
$(2,2)$-divisor, which we construct using wall-crossings. In the case where the
divisor is proportional to the anti-canonical divisor, the first author,
together with Ascher and Liu, developed a framework for wall crossings in
K-moduli and proved that there are only finitely many walls, which occur at
rational values of the coefficient $c$. This paper constructs the first example
of wall-crossing in K-moduli in the non-proportional setting, and we find a
wall at an irrational value of $c$. In particular, we obtain explicit
descriptions of the GIT and K-moduli spaces (for $c \leq 1/2$) of these
$(2,2)$-divisors. Furthermore, using the conic bundle structure, we study the
relationship with the GIT moduli space of plane quartic curves.
