Lifting Wilkie's theorem from $\mathbb{N}$ to other structures

Question

Let $\models,\models_2,\models_d$ be the satisfaction relations for first-order logic, second-order logic with full semantics, and second-order logic with set quantifiers ranging over definable-with-parameters sets respectively (note that this last one is generally different from Henkin semantics). Given a structure $\mathfrak{A}$, let $Th(\mathfrak{A}),Th_2(\mathfrak{A})$ be the full theories of $\mathfrak{A}$ in the sense of first-order and full second-order logic respectively.

Given a language $\Sigma$, a structure $\mathfrak{A}$, and a first-order $\Sigma$-theory $A$, say that $A$ Wilkie-captures $\mathfrak{A}$ iff the following three conditions hold:

1. $\mathfrak{A}\models A$.

2. Whenever $T$ is a finite first-order $\Sigma$-theory, $\psi_1(X),...,\psi_n(X)$ are first-order $\Sigma$-formulas with a free set variable $X$, and $T\cup\{\forall X\psi_i(X): 1\le i\le n\}\models_2Th_2(\mathfrak{A})$, then there is a finite first-order $\Sigma$-theory $S$ such that $S\cup\{\forall X\psi_i(X):1\le i\le n\}\models_d A$.

3. $A$ is "maximal-mod-finite" with the above properties: if $B$ is any other $\Sigma$-theory satisfying condition (1) and (2), then there is a finite set of first-order sentences $U$ such that $A\cup U\models B$.

(Note that if such a theory exists, it is only "mod-finite" unique; the natural setting here is a quotient of a substructure of the usual Lindenbaum algebra over the empty theory in $\Sigma$, namely the subsets of $Th(\mathfrak{A})$ partially preordered by $B_1\le B_2\leftrightarrow B_2\cup F_2\vdash B_1\cup F_1$ for some finite $F_1,F_2\subseteq Th(\mathfrak{A})$. I'm not very familiar with this structure, however.)

Wilkie proved that $\mathsf{PA}$ Wilkie-captures $\mathfrak{N}=(\mathbb{N};+,\times)$ (see section $7.4.1$ of Kossak/Schmerl, The structure of models of Peano arithmetic). I'm curious about the situation with respect to other structures. To keep things concrete (relatively), work in $\mathsf{ZFC}$ + large cardinals and let $\mathfrak{V}=(V_\kappa;\in)$ where $\kappa$ is the least strongly inaccessible.

Question: Is $\mathfrak{V}$ Wilkie-capturable? If so, is there a c.e. theory which Wilkie-captures $\mathfrak{V}$?

The proof of Wilkie's theorem seems rather particular to $\mathsf{PA}$, and while there are a number of similarities between set theories and theories of arithmetic I'm not very familiar with tools for transferring arguments between them.