GK-dimension of birationally commutative surfaces

Let $k$ be an algebraically closed field, let $K/k$ be a finitely generated field extension of transcendence degree $2$, let $\sigma \in \operatorname{Aut}_k(K)$, and let $A \subseteq Q = K[t; \sigma]$ be an $\mathbb{N}$-graded subalgebra with $\dim_k A_n < \infty$ for all $n \geq 0$. Then if $A$ is big enough in $Q$ in an appropriate sense, we prove that $\operatorname{GK} A = 3,4,5,$ or $\infty$, with the exact value depending only on the geometric properties of $\sigma$. The proof uses techniques in the birational geometry of surfaces which are of independent interest.