I was wondering if there is a characterization of "$\alpha$ is admissible" as a predicate, as I haven't come across such before. By definition, $\alpha$ is admissible iff $L_\alpha\vDash\Delta_0$-collection, but this doesn't actually give a formula for "$\alpha$ is admissible" because $\Delta_0$-collection is an axiom schema.
Now, it seems that the only way to formalize this would be to replace $\Delta_0$-collection (or equivalently $\Sigma_1$-collection) with a finite number of its instances, but I have never seen how this can be done.
You're wrong, because the statement $L_\alpha\models T$ is an internal statement, rather than an external statement. Here $T$ is an object of the universe, not the meta-theory.
And internally, the theory of $\Delta_0$-collection axioms is perfectly definable in the language of set theory, albeit via a very complicated formula. So stating that $\alpha$ is admissible if and only if $L_\alpha$ satisfies a certain theory, is in fact a single formula in the language of set theory.
In particular, Richter and Aczel's Inductive Definitions and Reflecting Properties of Admissible Ordinals (theorem 2.4) has a sentence $\sigma_0$ which is $\Pi_3$, where $L_\alpha\vDash\sigma_0$ iff $\alpha$ is admissible.