Categories of bounded $(\mathfrak{sp}(\mathrm S^2V\oplus\mathrm S^2V^*),\mathfrak{gl}(V))$- and $(\mathfrak{sp}(\Lambda^2V\oplus\Lambda^2V^*),\mathfrak{gl}(V))$-modules

Let $\mathfrak g$ be a reductive Lie algebra over $\mathbb C$ and $\mathfrak k\subset\mathfrak g$ be a reductive in $\mathfrak g$ subalgebra. We call a $\mathfrak g$-module $M$ a $(\mathfrak g,\mathfrak k)$-module whenever $M$ is a direct sum of finite-dimensional $\mathfrak k$-modules. We call a $(\mathfrak g,\mathfrak k)$-module $M$ bounded if there exists $C_M\in\mathbb Z_{\ge0}$ such that for any simple finite-dimensional $\mathfrak k$-module $E$ the dimension of the $E$-isotypic component is not more than $C_M\dim E$. Bounded $(\mathfrak g,\mathfrak k)$-modules form a subcategory of the category of $\mathfrak g$-modules. Let $V$ be a finite-dimensional vector space. We prove that the categories of bounded $(\mathfrak{sp}(\mathrm S^2V\oplus\mathrm S^2V^*),\mathfrak{gl}(V))$-modules and $(\mathfrak{sp}(\Lambda^2V\oplus\Lambda^2V^*),\mathfrak{gl}(V))$-modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of $V$.