Edit: When I was writing this post, I actually changed my mind in the middle about how I wanted to do this, so I think it came out a bit unfocused. The idea is straightforward. Fix a contravariant pointed functor $G$. We want to check that the natural map from $F\otimes^h G$ (unpointed derived tensor product) to $F\otimes^h_* G$ (pointed derived tensor product) is an equivalence for functors $F$ of the form $F=I_+\wedge\hom(x_0, -)$. This is good enough, because all other homotopy types of pointed functors can be built as repeated homotopy pushouts of functors of this type. So, I need to calculate both the pointed and unpointed derived tensor products of $F$ and $G$ for this type of $F$. The pointed tensor product is easy, because $F$ is cofibrant in the pointed models structure, so the derived product is equivalent to the strict product, which can be calculated using the YL. The unpointed tensor product is slightly less obvious, because it is not clear that $F$ is cofibrant in the unpointed model structure, and this is why the derived case does not follow immediately from the strict case. But, $F$ can be presented as a homotopy pushout of free (in the unpointed sense!) functors, and an elementary little calculation shows that the unpointed derived tensor product agrees with the pointed one.

This is a proof by calculation. Since the "calculation" is extremely easy, I feel it is not too bad. But it would be nice to see a conceptual reason why it ought to be true. I believe such a reason exists, but I have not been able to nail it down.

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I think the answer is yes. Here is an attempt at an argument.

Let $SS_*$ and $SS$ be the categories of pointed and unpointed simplicial sets. Let $[C, SS]$ be the category of all functors from $C$ to $SS$ and let $[C, SS_*]_*$ be the category of all pointed functors from $C$ to $SS_*$. Define similarly the functor categories $[C,Sets]$ and $[C, Sets_*]_*$. Consider $[C, Sets]$ and $[C,Sets_*]_*$ to be subcategories of $[C, SS]$ and $[C, SS_*]_*$ respectively.

The functor categories $[C, SS]$ and $[C, SS_*]_*$ have well-known model structures where weak equivalences and fibrations are defined pointwise. It is not difficult to describe the cofibrations explicitly. The cofibration in $[C, SS]$ are generated by maps of the form $$I\times \hom(x_0, -) \longrightarrow J\times \hom(x_0, -)$$ where $I, J$ are simplicial sets, $x_0$ is an object of $C$, $\hom(x_0, x)$ denotes the (pointed) set of morphisms in $C$, and the map is induced from a cofibration of simplicial sets $I\hookrightarrow J$. Similarly, the cofibrations in $[C, SS_*]_*$ are generated by maps of the form $$I_+\wedge \hom(x_0, -) \longrightarrow J_+\wedge \hom(x_0, -).$$

One can define homotopy tensor product using cofibrant replacement in this model structure. Namely, if F and G are two functors (either pointed or unpointed), then $B(G, C, F)\simeq cG \otimes cF$, where $c$ denotes a cofibrant replacement in the appropriate functor category. In fact, it is enough to take a cofibrant replacement of either $F$ or $G$. That is, $cG\otimes F\simeq G\otimes cF\simeq cG\otimes cF$.

There is an obvious forgetful functor that I will denote by $R$. $$R\colon [C, SS_*]_* \longrightarrow [C, SS].$$ Your question is equivalent to the following: does $R$ preserve homotopy coends? You only ask the question for set-valued functors, but I think the answer is yes in general. Let me formulate it a little more precisely. Let $F\colon C\to SS_*$ and $G\colon C^{op}\to SS_*$ be pointed functors. There is an evident natural map from the (unpointed) homotopy coend $RG\otimes^h RF$ to the pointed homotopy coend $G\otimes^h F$. We want to show that this map is an equivalence. Let us first check it when $F$ has the form $F(-)=I_+\wedge \hom(x_0, -)$ for some simplicial set $I$ and object $x_0$ of $C$. In this case, $F$ is cofibrant in $[C, SS_*]_*$, so the pointed homotopy coend of $F$ and $G$ is equivalent to the pointed strict coend which, by Yoneda Lemma, is equivalent to $I_+\wedge G(x_0)$. Now let us consider $RF$ and $RG$. It is not immediately obvious whether $RF$ is cofibrant in $[C, SS]$. On the other hand, $RF$ is objectwise equivalent to the following homotopy pushout $$*\times \hom(0, -)\longleftarrow I\times \hom(0, -) \longrightarrow I\times \hom(x_0, -) .$$ Taking homotopy coend with $RG$ preserves objectwise homotopy pushouts. It follows that $RF \otimes^h RG$ is equivalent to the following homotopy pushout $$*\times \hom(0, -)\otimes^h RG\longleftarrow I\times \hom(0, -)\otimes^h RG\longrightarrow I\times\hom(x_0, -) \otimes^h RG.$$ Which, again using Yoneda Lemma, together with the fact that $RG(0)=*$, implies that $RF \otimes^h RG$ is equivalent to $I_+\wedge G(x_0)$. So we obtain that $RF\otimes^h RG$ is equivalent to $F\otimes^h G$. With a little more careful diagram-chasing it should not be hard to see that the canonical map $RF\otimes^h RG\longrightarrow F\otimes^h G$ induces this equivalence.

For a general pointed $F$, one can present $F$ as a homotopy colimit along generating cofibrations in $[C, SS_*]_*$ (take a cofibrant replacement of $F$), and one obtains the result using a similar calculation plus induction.