I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state the question for the category of simplicial vector spaces $\mathsf{SVec}$.

As far as I understand, the structure of simplicial category for this kind of abelian categories of modules should be the same as the one for simplicial abelian groups from Goerss, Jardine, Theorem III.2.13. For $\mathsf{SVec}$:

The copowering functor $\cdot \otimes \cdot: \mathsf{SVec} \otimes \mathsf{SSet} \to \mathsf{SVec}$ is defined as $V \otimes K = V \otimes \mathbb{R} K$, the tensor product with the free simplicial vector space generated by $K$.

Then this fits in the two-variable adjunction

$$ \mathsf{SVec}(V \otimes K, W) \cong \mathsf{SSet} (K, \mathbf{Hom}_{\mathsf{SVec}}(V,W)) \cong \mathsf{SVec}(V, \mathbf{Hom}_{\mathsf{SSet}}(K, W)), \tag{1} $$

for any $K \in \mathsf{SSet}$, and $V,W \in \mathsf{SVec}$, where $\mathbf{Hom_{\mathsf{SVec}}}(V,W)$ is the mapping space between $V$ and $W$ (it consists of *linear* homotopies), and $\mathbf{Hom}_{\mathsf{SSet}}(K,W)$ is the mapping space between $K$ and the underlying simplicial set of $W$, which is actually the same as $\mathbf{Hom_{\mathsf{SVec}}}(\mathbb{R}K, W)$. Both of these also happen to be simplicial vector spaces and not just simplicial sets, but note that in the case $K=V$, they do not coincide.
Now I want to know is if $\mathsf{SVec}$ is closed monoidal with respect to $\otimes$ and $\mathbf{Hom_{\mathsf{SVec}}}$, that is: Is there a two-variable tensor-hom adjunction
$$ \mathsf{SVec}(V \otimes U, W) \cong \mathsf{SVec} (U, \mathbf{Hom}_{\mathsf{SVec}}(V,W)) \cong \mathsf{SVec}(V, \mathbf{Hom}_{\mathsf{SVec}}(U, W)), \tag{2} $$
for any $U,V,W \in \mathsf{SVec}$?

I managed to show that assuming that one can always choose a basis for a simplicial vector space, and write $U = \mathbb{R} B$, for some simplicial set $B$, there is a choice-dependent isomorphism as in (2) following from (1) since $\mathbf{Hom}_{\mathsf{SSet}}(B,W) = \mathbf{Hom}_{\mathsf{SVec}}(U,W)$. But is there a way to get a natural isomorphism?

In particular, is this adjunction general enough that it holds for all categories of simplicial $R$-modules over a ring $R$? If not, what are the minimal hypotheses for this to hold?

It seems to me this should be an example/exercise in books such as Hovey or Goerss, Jardine, but I could not find it there, so either it is more difficult than it looks or it is simply out of their main focus.