Unless I am very mistaken, there is an easy way to establish a bound of the
"much better" kind mentioned in the comments above. (I don't doubt one can and should give a more precise answer.)

Write $\phi$ for $\pi \sigma \pi^{-1}$
(meaning $\pi^{-1}\circ \sigma \circ \pi$),
where $\sigma = (1 2 \dotsc n)$. Then
$\delta(\pi(S))$ is the number of $m\in S$ such that $\phi(m) \notin S$.

(Well, there could be a difference of $1$, which does not matter. Let us redefine $\delta(S)$ to be the number of $m\in S$ such that $\sigma(m)\notin S$. Perhaps we should consult OEIS again, after this redefinition?)

For the sake of simplicity, we shall restrict our attention to maps $\pi$
without fixed points; that implies $\phi$ has no fixed points either.
(This case covers the case of $\pi$ ranging over $n$-cycles, in particular.)
Let $v=v_\phi$ be the number of $m\in [n]$ such that
$\phi(m) \notin \{\sigma(m), \sigma^{-1}(m)\}$.
A bit of doodling shows that
the number $w=w_\phi$ of consecutive pairs $A=\{m,\sigma(m)\}$ such that
$\phi(A)\cap A=\emptyset$ (called them "valid" pairs) is $\geq v_\phi/2$.

Now we build a set $S$ as follows. At each step, we add to $S$ a valid pair
$\{m,\sigma(m)\}$ that has not yet been marked as forbidden. We also
mark eight pairs as forbidden:
$$\{\phi(m),\sigma^{\pm 1}(\phi(m))\},
\{\phi(\sigma(m)),\sigma^{\pm 1}(\phi(\sigma(m)))\},
\{\phi^{-1}(m),\sigma^{\pm 1}(\phi^{-1}(m))\},
\{\phi^{-1}(\sigma(m)),\sigma^{\pm 1}(\phi^{-1}(\sigma(m)))\}.$$
In this way, we get to build a set $S$ of size at least
$2 \cdot w_\phi/9 \geq v_\phi/9$ with $\delta(S)\leq |S|/2$ and
$\phi(S)\cap S = \emptyset$, so that $\delta(\pi(S)) = |S|$.
Hence
$$D(\pi) \geq v_\phi/18.$$

Thus, the number of distinct $\phi$ without fixed points
coming from permutations $\pi$ with $D(\pi)\leq k$ is at most
$2^n n^{18 k}$ (in fact, at most $2^{n-18k} n^{18 k}$). Now,
at most $n$ permutations $\pi$ (in fact, exactly $n$ permutations $\pi$) give
rise to the same $\phi = \pi \sigma \pi^{-1}$. Therefore, the total number
of permutations without fixed points
$\pi$ of $[n]$ such that $D(\pi)\leq k$ is at most
$$2^n n^{18 k + 1}.$$

I think (though I haven't checked yet) that the analysis for arbitrary
permutations $\pi$ should require only a little more work.

5more comments