We use a method developed by Björklund and Gorodnik to show a central limit theorem (as $T$ tends to $\infty$) for the counting functions $\# \left( Λ\cap Ω_T \right)$ where $Λ$ ranges over the space $Y_{2d}$ of symplectic lattices in $\mathbb{R}^{2d}$ ($d \geqslant 4$). Here $\lbrace Ω_T \rbrace_T$ is a certain family of bounded domains in $\mathbb{R}^{2d}$ that can be tessellated by means of the action of a diagonal semigroup contained in $\mathrm{Sp}(2d, \mathbb{R})$. In the process we obtain new $L^p$ bounds on a certain height function on $Y_{2d}$ originally introduced by Schmidt.