This is not an answer, but it is a bit too long for a comment. And perhaps it provides some hint towards an actual solution. Basically, I have a counterexample where the monoidal structure of $V$ is not cartesian, but I don't see if it can be modified to make it cartesian.

Let $V$ be the category of $\mathbb{Z}/2$-[graded sets][1]. The category $C_0 = \mathsf{Cat}$ has a nice $V$-enrichment where even morphisms are covariant functors and odd morphisms are contravariant functors. It also has (co)tensors, the tensor of a category $X$ by a graded set $A$ is $X \times A_0 \sqcup X^{\mathrm{op}} \times A_1$ and the cotensor is $X^{A_0} \times (X^{\mathrm{op}})^{A_1}$. However, the internal homs in $C_0$ (with respect to the cartesian monoidal structure) don't lift to $C$. Indeed, the odd elements of $C(X \times Y, Z)$ are functors that are contravariant in both $X$ and $Y$, but the odd elements of $C(X, Z^Y)$ are functors that are contravariant only in $X$.

  [1]: https://ncatlab.org/nlab/show/graded+set