I am not an expert of this area and I put it as an answer rahter than a comment just because it's too long.

I think an important work on this topic is "The homotopy theory of dg-categories and derived morita theory" by Bertrand Toen. He uses dg-categories instead of $\infty$-categories, so does the paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry".

One of the main result (Corollary 4.8) in this paper is: If $\mathcal{C}$ and $\mathcal{D}$ are two small dg-categories, then there is a functorial bijection between the set of maps $[\mathcal{C},\mathcal{D}]$ in $Ho(dg-Cat)$, the homotopy category of small dg-categories, and the set of isormorphism classes of right quasi-representable objects in $Ho((\mathcal{C}\otimes \mathcal{D}^{op})-Mod)$, the homotopy category of $\mathcal{C}-\mathcal{D}$ bi-modules.

The $\mathcal{C}-\mathcal{D}$ bi-modules, rather than $\mathcal{C}\otimes \mathcal{D}$, is natural to give functors from $\mathcal{C}$ to $\mathcal{D}$. Hence as Will Sawin comments above, you may need a self-dual statement to get the isomorphism you want.

By the way, the isomorphism $$ QC(X\times Y)\cong Fun^L(QC(X),QC(Y)) $$ is a (not very straightforward) consequence of the above result, which is Theorem 8.9 in Toen's paper, where he checks the self-duality of $QC(X)$.

Maybe you could investigate your categories $\mathcal{A}$ and $\mathcal{B}$ to see whether they fit into Toen's framework.

I am not an expert of this area and I put it as an answer rahter than a comment just because it's too long.

I think an important work on this topic is "The homotopy theory of dg-categories and derived morita theory" by Bertrand Toen. He uses dg-categories instead of $\infty$-categories, so does the paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry".

One of the main result (Corollary 4.8) in this paper is: If $\mathcal{C}$ and $\mathcal{D}$ are two small dg-categories, then there is a functorial bijection between the set of maps $[\mathcal{C},\mathcal{D}]$ in $Ho(dg-Cat)$, the homotopy category of small dg-categories, and the set of isormorphism classes of right quasi-representable objects in $Ho((\mathcal{C}\otimes \mathcal{D}^{op})-Mod)$, the homotopy category of $\mathcal{C}-\mathcal{D}$ bi-modules.

The (right quasi-representable) $\mathcal{C}-\mathcal{D}$ bi-modules, rather than $\mathcal{C}\otimes \mathcal{D}$, is natural to give functors from $\mathcal{C}$ to $\mathcal{D}$. Hence as Will Sawin comments above, you may need a self-dual statement to get the isomorphism you want.

By the way, the isomorphism $$ QC(X\times Y)\cong Fun^L(QC(X),QC(Y)) $$ is a (not very straightforward) consequence of the above result, which is Theorem 8.9 in Toen's paper, where he checks the self-duality of $QC(X)$.

Maybe you could investigate your categories $\mathcal{A}$ and $\mathcal{B}$ to see whether they fit into Toen's framework.