Structure of a least-order planar counterexample

Let $V$ be a type in a fixed universe with a finite-type structure, let $G$ be a simple graph on $V$, and let $D$ be a binary relation on $V$. Assume that there is an injective map $p:V\to\mathbb R^2$ such that vertices not incident with an edge do not lie on that edge’s closed straight segment, and such that the closed segments representing any two edges with distinct endpoints are disjoint; assume also that $D$ has no loops, every arc of $D$ follows an edge of $G$, and every edge of $G$ is oriented in exactly one direction by $D$. Assume further that there is no coloring $c:V\to\mathrm{Fin}\,2$ for which, for each of the two colors $k$, no list of length at least $3$ with distinct vertices, directed arcs between consecutive entries, and a directed arc from its last entry back to its first consists entirely of vertices colored $k$. Finally, assume that for every finite type $W$ in the same universe, every simple graph $H$ on $W$, and every relation $E$ on $W$, if $H$ has an embedding satisfying the same planarity conditions, $E$ orients every edge of $H$ in exactly one direction without loops, and $|W|<|V|$, then such a coloring exists for $E$. Under these assumptions, $V$ is nonempty, every ordered pair of vertices $u,v\in V$ is connected by a directed walk from $u$ to $v$ of zero or more arcs, and every vertex of $G$ has at least three neighbors.