OK, so I wanted to elaborate here on Sam's helpful comments.

Set $N = 8g-4$. An explicit description of the $N$-gon $F$ that I have in mind is \begin{equation} F = D \ \backslash \ \bigcup_{j=1}^{N} \left(\sqrt{a-1} \cdot D + \sqrt{a} e^{2\pi i(j-2g)/N}\right) \end{equation} with $a = \sec \frac{2\pi}{N}$. In particular, the nearest point to the origin is at a Euclidean distance $u := \sqrt{a} - \sqrt{a-1}$, so the hyperbolic distance is $d = \int_0^u \frac{dr}{1-r^2} = \frac{1}{2}\log\frac{1+u}{1-u}$, which evidently grows as $\log g$.

A bit more context also: I expect that $t_g f(g) \approx const$, where $t_g$ is whatever timescale and $f(g)$ is the quantity mentioned in the question. If $t_g \sim \log g$ then I'd expect that $f(g) \sim 1/\log g$, which is actually a weak enough dependence on $g$ to not be surprising based on the numerics alluded to in the question.