More specifically, let $S_k$ be the spin representation of $\mathrm{Spin}(2k+1)$.
Then is there are action of $\mathrm{Sp}(2r-2)$ on $\otimes^{2r}S_k$ which commutes
with the action of $\mathrm{Spin}(2k+1)$?

I am not asking for a duality. That is, I do not require that each isotypic vector
space is an irreducible representation of $\mathrm{Sp}(2r-2)$.

The motivation is that the space of invariant tensors in $\otimes^{2r}S_k$ looks like
the irreducible representation of $\mathrm{Sp}(2r-2)$ with highest weight $k\omega_{r-1}$.
For instance, for $k=1$, both vector spaces have dimension given by the Catalan numbers $C_r = \frac{1}{r+1}\binom{2r}{r}$. More generally, for arbitrary $k$ the dimension of both of these spaces is
$$ \prod_{1\leq i \leq j \leq r-1} \frac{i+j+2k}{i+j} $$
which is the number of so-called "*$k$-fans of nested Dyck paths of semilength $r$*" (see, e.g., https://arxiv.org/abs/1009.4690).