Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete pretriangulated dg-categories, or equivalently dg-categories for which the canonical functor $T \hookrightarrow D^c(T)$, into the dg-category of compact dg-modules, is an equivalence. Its dualizable objects are precisely the saturated dg-categories (Cisinski-Tabuada, Theorem 5.8). These correspond to schemes over $k$ which are smooth and proper.

Let $A$ and $B$ be small dg-categories and consider To"en's formula $$Fun_k(A,B) = D_{rqr}(A^{op} \otimes B)$$ for the internal hom in the $\infty$-category of dg-categories, where $D_{rqr}$ is the dg-category of right quasi-representable bimodules. If $A$ and $B$ are saturated, this reduces to an equivalence of dg-categories $$Fun_k(A,B) = D_c(A^{op} \otimes B)$$ where $D_c$ is the dg-category of compact dg-modules. If $A$ and $B$ are Morita fibrant, and $A$ has the property of self-duality, then this gives an equivalence of dg-categories $$ Fun_k(A, B) = A \otimes B. $$ It is clear that this recovers the integral transform formula for perfect complexes, in the case of smooth proper $k$-schemes.

This is only the small version of your question. In the presentable case all I can say at the moment is the following. If $A$ and $B$ are locally presentable dg-categories, compactly generated with $A_c$ and $B_c$ the full subcategories of compact objects, then $$ Fun^L_k(A,B) = Fun^L_k(\hat{A_c}, \hat{B_c}) = (A_c^{op} \otimes B_c)^{\wedge} = (A_c \otimes B_c)^{\wedge} $$ where the second equivalences uses Corollary 7.6 in To"en's "Derived Morita theory". Here $(-)^\wedge$ denotes the free cocompletion, and I am assuming self-duality for $A$ again. This seems to recover the integral transform formula in the case of schemes, using $Perf(X \times_k Y) = Perf(X) \otimes Perf(Y)$.

(It has been a while since I looked at this stuff, so let me know if I'm making a mistake somewhere.)

Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete dg-categories, or equivalently dg-categories for which the canonical functor $T \hookrightarrow D^c(T)$, into the dg-category of compact dg-modules, is an equivalence. Its dualizable objects are precisely the saturated dg-categories (Cisinski-Tabuada, Theorem 5.8). These correspond to schemes over $k$ which are smooth and proper.

Let $A$ and $B$ be small dg-categories and consider To"en's formula $$Fun_k(A,B) = D_{rqr}(A^{op} \otimes B)$$ for the internal hom in the $\infty$-category of dg-categories, where $D_{rqr}$ is the dg-category of right quasi-representable bimodules. If $A$ and $B$ are saturated, this reduces to an equivalence of dg-categories $$Fun_k(A,B) = D_c(A^{op} \otimes B)$$ where $D_c$ is the dg-category of compact dg-modules. If $A$ and $B$ are idempotent complete, and $A$ has the property of self-duality, then this gives an equivalence of dg-categories $$ Fun_k(A, B) = A \otimes B. $$ It is clear that this recovers the integral transform formula for perfect complexes, in the case of smooth proper $k$-schemes.

This is only the small version of your question. In the presentable case all I can say at the moment is the following. If $A$ and $B$ are locally presentable dg-categories, compactly generated with $A_c$ and $B_c$ the full subcategories of compact objects, then $$ Fun^L_k(A,B) = Fun^L_k(\hat{A_c}, \hat{B_c}) = (A_c^{op} \otimes B_c)^{\wedge} = (A_c \otimes B_c)^{\wedge} $$ where the second equivalences uses Corollary 7.6 in To"en's "Derived Morita theory". Here $(-)^\wedge$ denotes the free cocompletion, and I am assuming self-duality for $A$ again. This seems to recover the integral transform formula in the case of schemes, using $Perf(X \times_k Y) = Perf(X) \otimes Perf(Y)$.

(It has been a while since I looked at this stuff, so let me know if I'm making a mistake somewhere.)

Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete pretriangulated dg-categories, or equivalently dg-categories for which the canonical functor $T \hookrightarrow D^c(T)$, into the dg-category of compact dg-modules, is an equivalence. Its dualizable objects are precisely the saturated dg-categories (Cisinski-Tabuada, Theorem 5.8). These correspond to schemes over $k$ which are smooth and proper.

Let $A$ and $B$ be small dg-categories and consider To"en's formula $$Fun_k(A,B) = D_{rqr}(A^{op} \otimes B)$$ for the internal hom in the $\infty$-category of dg-categories, where $D_{rqr}$ is the dg-category of right quasi-representable bimodules. If $A$ and $B$ are saturated, this reduces to an equivalence of dg-categories $$Fun_k(A,B) = D_c(A^{op} \otimes B)$$ where $D_c$ is the dg-category of compact dg-modules. If $A$ and $B$ are Morita fibrant, and $A$ has the property of self-duality, then this gives an equivalence of dg-categories $$ Fun_k(A, B) = A \otimes B. $$ It is clear that this recovers the integral transform formula for perfect complexes, in the case of smooth proper $k$-schemes.

This is only the small version of your question. In the presentable case all I can say at the moment is the following. If $A$ and $B$ are locally presentable dg-categories, compactly generated with $A_c$ and $B_c$ (the full subcategories of compact objects) both saturated, then $$ Fun^L_k(A,B) = Fun^L_k(\hat{A_c}, \hat{B_c}) = (A_c^{op} \otimes B_c)^{\wedge} = (A_c \otimes B_c)^{\wedge} $$ where the second equivalences uses Corollary 7.6 in To"en's "Derived Morita theory". Here $(-)^\wedge$ denotes the free cocompletion, and I am assuming self-duality for $A$ again.

(It has been a while since I looked at this stuff, so let me know if I'm making a mistake somewhere.)

Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete pretriangulated dg-categories, or equivalently dg-categories for which the canonical functor $T \hookrightarrow D^c(T)$, into the dg-category of compact dg-modules, is an equivalence. Its dualizable objects are precisely the saturated dg-categories (Cisinski-Tabuada, Theorem 5.8). These correspond to schemes over $k$ which are smooth and proper.

Let $A$ and $B$ be small dg-categories and consider To"en's formula $$Fun_k(A,B) = D_{rqr}(A^{op} \otimes B)$$ for the internal hom in the $\infty$-category of dg-categories, where $D_{rqr}$ is the dg-category of right quasi-representable bimodules. If $A$ and $B$ are saturated, this reduces to an equivalence of dg-categories $$Fun_k(A,B) = D_c(A^{op} \otimes B)$$ where $D_c$ is the dg-category of compact dg-modules. If $A$ and $B$ are Morita fibrant, and $A$ has the property of self-duality, then this gives an equivalence of dg-categories $$ Fun_k(A, B) = A \otimes B. $$ It is clear that this recovers the integral transform formula for perfect complexes, in the case of smooth proper $k$-schemes.

This is only the small version of your question. In the presentable case all I can say at the moment is the following. If $A$ and $B$ are locally presentable dg-categories, compactly generated with $A_c$ and $B_c$ the full subcategories of compact objects, then $$ Fun^L_k(A,B) = Fun^L_k(\hat{A_c}, \hat{B_c}) = (A_c^{op} \otimes B_c)^{\wedge} = (A_c \otimes B_c)^{\wedge} $$ where the second equivalences uses Corollary 7.6 in To"en's "Derived Morita theory". Here $(-)^\wedge$ denotes the free cocompletion, and I am assuming self-duality for $A$ again. This seems to recover the integral transform formula in the case of schemes, using $Perf(X \times_k Y) = Perf(X) \otimes Perf(Y)$.

(It has been a while since I looked at this stuff, so let me know if I'm making a mistake somewhere.)