$\def\Fin{\text{Fin}_*} \def\Set{\text{Set}_*} \def\dd{\mathop{\diamond_\land}}$

The present question is intimately related to another question.

Let $\Fin$ be the category of pointed sets.

The category $[\Fin,\Set]$ of $\Set$-enriched functors has a convolution monoidal product when both categories are endowed with the smash product operation.

This means that we can convolve two parallel functors $F,G : \Fin \to \Set$ to $$ F*G := n\mapsto \int^{pq}Fp\land Gq\land \Fin(p\land q,n) $$ One would be tempted to define, starting from this, a substitution product following Kelly's "On the operads of PJ May": the iterated convolution $$ F^{\ast m} \cong \int^{p_1,...,p_m} Fp_1\land \dots \land Fp_m \land \Fin(p_1\land\dots\land p_m,\_) $$ should define a substitution product $$ F \dd G := n\mapsto \int^m Fm\land G^{\ast m}n $$ However, this is not an associative operation: in Kelly's proof, the fact that convolution and substitution interact in the following way is crucial: $$ (F\dd G)^{\ast m} \cong F^{\ast m}\dd G\tag{$\star$} $$ Because once $(\star)$ is proved, it follows that $$ \begin{align*} (F\dd G)\dd H & = \int^n (F\dd G)n\land H^{*n} \\ & \cong \int^{nm} Fm\land G^{*m}n\land H^{*n} \\ F \dd (G \dd H) & = \int^m Fm \land (G\dd H)^{*m} \\ & \cong \int^m Fm \land (G^{*m}\dd H) \\ & \cong \int^{nm} Fm\land G^{*m}n\land H^{*n}. \end{align*} $$

Let's try to prove $(\star)$ then, adopting a slightly improper notation in order to save space: $\underline p$ is a tuple of the appropriate length, and we write $\int^{\underline{p}} F\underline p \land \Fin(\underline p^\land,\_)$ for the integral defining $F^{\ast m}$: $$ \begin{align*} (F\diamond_\land G)^{\ast m} & =\int^{\underline{k}} \Fin(\underline{k}^\land, \_)\land (F\diamond_\land G)(\underline{k}) \\ & \cong \int^{\underline{k}, \underline{p}} \Fin(\underline{k}^\land, \_) \land \left(Fp_1 \land G^{\ast p_1}k_1\right) \land \cdots \land \left(Fp_m \land G^{\ast p_m}k_m\right) \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land (G^{\ast p_1} \ast\cdots\ast G^{\ast p_m}) \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land G^{\ast \sum p_i} \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land \int^r G^{\ast r}\land \Fin(\sum p_i,r) \end{align*} $$ whre I expanded the definition, rearranged terms, compacted again the definition and finally used the fact that $(\_)^{\ast \_} : [\Fin,\Set]\times \Fin^\text{op} \to [\Fin,\Set]$ is a bifunctor, so that by ninja Yoneda $G^{\ast k} \cong \int^r G^{\ast r}\land \Fin(k,r)$.

Now I'm stuck, because $\Fin(\sum p_i,r) \not\cong \Fin(\bigwedge p_i,r) $; the monoidal structure given by coproduct is different from the one given by smash product; the first categorifies the monoid structure $(\mathbb N, +)$, whereas the second categorifies the monoid structure $(\mathbb N, \circ)$, where $p\circ q := pq-p-q+1$ (if $p,q$ are finite sets with $p,q$ elements; or counting in another way, if $p:= \{0,...,p\}$ then $p\circ q=pq$).

If I had obtained $\Fin(\bigwedge p_i,r)$ instead of $\Fin(\sum p_i,r)$, all would have been ok, because now $$ \begin{align*} & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land \int^r G^{\ast r}\land \Fin\big(\bigwedge p_m,r\big) \\ & \cong \int^{\underline{p},r} \Fin(\underline{p}^\land,r) \land Fp_1\land\cdots\land Fp_m \land G^{\ast r} \\ & \cong \int^r F^{\ast m} r \land G^{\ast r} \\ & \cong F^{\ast m} \diamond_\land G \end{align*} $$

It seems then that there is no way to define an operad-like monoidal structure on $[\Fin,\Set]$, neither starting from the framework of my previous question, nor in a purely $\Set$-enriched one, i.e. taking the smash-product monoidal structure on both domain and codomain of $[\Fin,\Set]$.

Why? What went wrong?