This is a try to enforce having standard models of ZFC in a way similar to how Omega rule do it for PA. I made a failed try at the posting titled "Omega rule and standard models of ZFC?", here I'll present it in a different way. It appears that the heart of ZFC is axiom of Foundation, now since $\in-$induction proves Regularity, then I'll try to make an Epsilon $\omega-$rule version of it [much as the omega rule for arithmetic can be viewed as strengthening the usual induction schema of PA].
$Epsilon \ \omega-rule$: if $\{\phi_1(y), \phi_2(y), \phi_3(y),...\}$ is the set of ALL formulas in the first order language of ZFC in which only symbol $``y"$ occur free, and only free, and symbol $``x"$ never occur, and if $\psi(y)$ is a formula in the same language in which only symbol $``y"$ occurs free, and only free, and symbol $``x"$ never occur, and if $\psi(x)$ is the formula obtained from $\psi(y)$ by merely replacing all occurrences of symbol $``y"$ by symbol $``x"$; then
from: $for \ i=1,2,3,.... \\ \forall x [\forall y (y \in x \leftrightarrow \phi_i(y)) \to (\forall y \in x (\psi(y))\to \psi(x))]$
we infer:
$\forall x \psi(x)$
In English: if we have every parameter free definable set fulfilling the antecedent of $\in-$induction for a parameter free definable property after formula $\psi$, then all sets would satisfy that property.
Question: would ZFC formulized in a language extended with the above rule, have all of its models being standard models?
The answer is still no, for the same reasons:
Any pointwise definable model of set theory satisfies your principle trivially since everything is parameter-freely definable (so your principle is just $\in$-induction for such models, hence follows from ZFC as usual), and there are ill-founded pointwise definable models.
More generally, what you've written is still a computable infinitary formula, so by the Barwise-Kreisel compactness theorem (discussed in more detail in my answer to your previous question) either it drastically restricts the "height" of the models which satisfy it or there are non-well-founded models which satisfy it.
However, let me go one step further with that second bulletpoint, and point out something that didn't occur to me in my answer to your previous question. Note that any well-founded model of ZFC has height at least $\omega_1^{CK}$ (this is a good exercise), so we can in fact use the BKCT to show that if $\theta$ is a computable infinitary sentence consistent with ZFC then ZFC+$\theta$ has ill-founded models (any well-founded model of ZFC+$\theta$ has "too large rank" for $\theta$ to avoid having ill-founded models). So there is a real sense in which anything along the lines you propose is doomed.