This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting $g(x)$ to $g(y)$ in the relative homotopy class of $g(\alpha)$, then $\ell_N(\beta) \leq \ell_\rho(\alpha)$. Similarly, if $\alpha$ is a geodesic loop in $S$ then the geodesic representative $\beta$, of $g(\alpha)$, has $\ell_N(\beta) \leq \ell_\rho(\alpha)$.

It follows that $R = R_\rho$, the injectivity radius of $(S, \rho)$, is greater than or equal to the injectivity radius of $N$. Let $x, y \in S$ be any pair of points. Let $\alpha$ be the shortest geodesic segment in $S$ connecting $x$ to $y$. For any point $z$ of $\alpha$ let $D \subset S$ be the open ball of radius $R$ about $z$. It follows that $\alpha \cap D$ is a single arc, centered at $z$. Thus the $R/2$ neighborhood of $\alpha$ is an embedded strip with area less than the area of $(S, \rho)$. Since the area of $(S, \rho)$ is $-2\pi\chi(S)$, deduce an upper bound for the length of $\alpha$ and hence for the diameter of $S$.

So -

Reading the third to last sentence of your post, I think that you may be asking a different question from what you actually wrote in the first half. That is, instead of the distances $d_N(g(x), g(y))$ and $d_\rho(x,y)$, you are interested in the lengths of certain geodesic arcs connecting the points.... Looking at Minsky's paper, I think that a version with $x \neq y$ is basically exactly the same compactness argument as his version with $x = y$.

This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting $g(x)$ to $g(y)$ in the relative homotopy class of $g(\alpha)$, then $\ell_N(\beta) \leq \ell_\rho(\alpha)$. Similarly, if $\alpha$ is a geodesic loop in $S$ then the geodesic representative $\beta$, of $g(\alpha)$, has $\ell_N(\beta) \leq \ell_\rho(\alpha)$.

It follows that $R = R_\rho$, the injectivity radius of $(S, \rho)$, is greater than or equal to the injectivity radius of $N$. Let $x, y \in S$ be any pair of points. Let $\alpha$ be the shortest geodesic segment in $S$ connecting $x$ to $y$. For any point $z$ of $\alpha$ let $D \subset S$ be the open ball of radius $R$ about $z$. It follows that $\alpha \cap D$ is a single arc, centered at $z$. Thus the $R/2$ neighborhood of $\alpha$ is an embedded strip with area less than the area of $(S, \rho)$. Since the area of $(S, \rho)$ is $-2\pi\chi(S)$, deduce an upper bound $A$ for the length of $\alpha$ and hence for the diameter of $S$.

So -

Reading the third to last sentence of your post, I think that you may be asking a different question from what you actually wrote in the first half. That is, instead of the distances $d_N(g(x), g(y))$ and $d_\rho(x,y)$, you are interested in upper bounds for certain geodesic arcs connecting the points.... Looking at Minsky's paper, I think that the proof with $x \neq y$ is basically the same compactness argument as his version with $x = y$.

This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting $g(x)$ to $g(y)$ in the relative homotopy class of $g(\alpha)$, then $\ell_N(\beta) \leq \ell_\rho(\alpha)$. Similarly, if $\alpha$ is a geodesic loop in $S$ then the geodesic representative $\beta$, of $g(\alpha)$, has $\ell_N(\beta) \leq \ell_\rho(\alpha)$.

It follows that $R = R_\rho$, the injectivity radius of $(S, \rho)$, is greater than or equal to the injectivity radius of $N$. Let $x, y \in S$ be any pair of points. Let $\alpha$ be the shortest geodesic segment in $S$ connecting $x$ to $y$. For any point $z$ of $\alpha$ let $D \subset S$ be the open ball of radius $R$ about $z$. It follows that $\alpha \cap D$ is a single arc, centered at $z$. Thus the $R/2$ neighborhood of $\alpha$ is an embedded strip with area less than the area of $(S, \rho)$. Since the area of $(S, \rho)$ is $-2\pi\chi(S)$, deduce an upper bound for the length of $\alpha$ and hence for the diameter of $S$.

So -

I think that you may be asking a different question from what you wrote? That is, instead of the quantities $d_N(g(x), g(y))$ and $d_\rho(x,y)$, you are interested in the lengths of certain geodesic arcs connecting these points....

This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting $g(x)$ to $g(y)$ in the relative homotopy class of $g(\alpha)$, then $\ell_N(\beta) \leq \ell_\rho(\alpha)$. Similarly, if $\alpha$ is a geodesic loop in $S$ then the geodesic representative $\beta$, of $g(\alpha)$, has $\ell_N(\beta) \leq \ell_\rho(\alpha)$.

It follows that $R = R_\rho$, the injectivity radius of $(S, \rho)$, is greater than or equal to the injectivity radius of $N$. Let $x, y \in S$ be any pair of points. Let $\alpha$ be the shortest geodesic segment in $S$ connecting $x$ to $y$. For any point $z$ of $\alpha$ let $D \subset S$ be the open ball of radius $R$ about $z$. It follows that $\alpha \cap D$ is a single arc, centered at $z$. Thus the $R/2$ neighborhood of $\alpha$ is an embedded strip with area less than the area of $(S, \rho)$. Since the area of $(S, \rho)$ is $-2\pi\chi(S)$, deduce an upper bound for the length of $\alpha$ and hence for the diameter of $S$.

So -

Reading the third to last sentence of your post, I think that you may be asking a different question from what you actually wrote in the first half. That is, instead of the distances $d_N(g(x), g(y))$ and $d_\rho(x,y)$, you are interested in the lengths of certain geodesic arcs connecting the points.... Looking at Minsky's paper, I think that a version with $x \neq y$ is basically exactly the same compactness argument as his version with $x = y$.