Possibly something like Separation by $\neg S$: $$\forall a \exists x \forall y (y \in x \leftrightarrow y \in a \land \neg Sy)$$ The idea is that if we define natural as: well founded transitive set of transitive sets, that when nonempty then it must have a predecessor, and such that every nonempty element of it must have a predecessor, then this separation would prevent having a natural that fulfills $\neg S$. Otherwise any such a natural $n$ would have a nonempty subset of it that is the set of all of its elements that fulfill $\neg S$, but that set wont have a minimal! Thus $n$ won't be well founded, which contradicts $n$ being a natural. So this theory would prove that any natural fulfills $S$, therefore by set construction proving the existence of the a of all naturals, which is an inductive set!.
However, I do suspect some kind of inconsistency lurking here or there?!
Regarding the consistency issue, I'd say that if the axiom of Heredity is weakened to two axioms:
Axiom of Heredity: $$\forall x (Sx \to \forall y \in x (Sy))$$
Axiom of Subsets: $$\forall x \forall y (Sx \land y \subset x \to Sy)$$
And if we weaken the Axiom of comprehension as to additionally require all parameters of $\phi$ to satisfy $S$.
Then the resulting theory plus Separation by $\neg S$ is a proper fragment of Muller's class theory! And so it's consistent relative to Muller's class theory.
So the question of consistency of addition of Separation by $\neg S$ to the axioms stated above, is mostly related to the upward heredity statement $$ \forall x (\forall y \in x (Sy) \to S(x))$$, and to unleashing parameters in comprehension. Those need to be checked.