Rational functions sharing preimages and height functions

Let $A$ and $B$ be non-constant rational functions over $\mathbb{C}$, and let $K \subset \mathbb{P}^1(\mathbb{C})$ be an infinite set. Using height functions, we prove that the inclusion $A^{-1}(K) \subseteq B^{-1}(K)$ implies the inequality ${\rm deg} B \geq {\rm deg} A$ in the following two cases: the set $K$ is contained in $\mathbb{P}^1(k)$, where $k$ is a finitely generated subfield of $\mathbb{C}$, or the set $K$ is discrete in $\mathbb{C}$, and $A$ and $B$ are polynomials. In particular, this implies that for $A$, $B$, and $K$ as above, the equality $A^{-1}(K) = B^{-1}(K)$ is impossible, unless ${\rm deg} B = {\rm deg} A$.