In this paper, Mathias also includes the axiom of infinity and the powerset axiom in $\operatorname{KP}^{\mathcal{P}}$. Infinity is fine with me, as it will always hold in the structures I am interested in. Also, since all my models are transitive, I have full foundation for free. However, the powerset axiom typically fails. Including the powerset axiom in $\operatorname{KP}^{\mathcal{P}}$ seems curious to me, as - on first sight - it appears to render the additional predicate $\mathcal{P}$ rather pointless... I'll think about it for a little while.

@Asaf Yes, I'm aware. However, since $x \in \mathcal{N}$ is not a set of ordinals in Ralf's book, this seemed more relevant to OP. The underlying reason though, that $L[x] = L(x)$, is - as you hinted - that it can be recursively coded as a subset of $\omega$.

Also note that $L[x]$ is not the least inner model containing $x$ - the least inner model containing $x$ is $L(x)$. $L[x]$, in contrast, is the least inner model closed under $y \mapsto y \cap x$. However, for $x \in \mathcal{N}$, these notions agree, i.e. $L[x]= L(x)$.