Subcanonical topology for Grothendieck quasitopos

Question

Let $\mathbb{C}$ be a small category equipped with two Grothendieck topologies $J, K$ with $J \subseteq K$ (i.e. $(\mathbb{C}, J, K)$ is a bisite), and consider the Grothendieck quasitopos $\mathsf{BiSep}(\mathbb{C}, J, K)$, which is the full subcategory of $\mathsf{Set}^{\mathbb{C}^{\mathsf{op}}}$ consisting of the presheaves that are sheaves for $J$ and separated for $K$. It is known that any Grothendieck topos of sheaves is equivalent to a topos of sheaves on a subcanonical site; is the same true for the Grothendieck quasitopos $\mathsf{BiSep}(\mathbb{C}, J, K)$? I.e. is there another bisite $(\mathbb{D}, J', K')$ with $J'$ subcanonical and $\mathsf{BiSep}(\mathbb{C}, J, K) \simeq \mathsf{BiSep}(\mathbb{D}, J', K')$, and if so, is there an explicit description of this bisite $(\mathbb{D}, J', K')$?