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](http://www.math.univ-toulouse.fr/~dcisinsk/MonoidalMotives5.pdf), 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.)