Let $\mathsf{KPh}$ denote the theory $\mathsf{KP}+``\textrm{The recursively inaccessibles are unbounded}\! "$. I haven't found an explicit analysis of $\mathsf{KPh}$ in Rathjen's preprints, but there are some implicit and non-detailed claims (e.g. in "Proof theory: From arithmetic to set theory") that the techniques used to analyze $\mathsf{KPi}$ ("admissible proof theory" techniques) carry over to analyzing $\mathsf{KPh}$.
If you accept an answer justified only by conjecture, in Taranovsky's ordinal notation "Degrees of Reflection" (Taranovsky, "Ordinal Notation", section 4.2), it's claimed that when $F$ is some natural property, the proof-theoretic ordinal of $\mathsf{KP}+``\textrm{The universe is }F\! "$ is often $C(C(\Omega,a),0)$, where $a$ is the term assigned to $\textrm{min}\{\alpha\mid L_\alpha\vDash F\}$ - I believe this is either a typo of $\textrm{min}\{\alpha\mid L_\alpha\vDash\mathsf{KP}+F\}$ or a weak claim, since if $F$ is a condition like "the admissibles are unbounded" then $\mathsf{KP}+F$ is already as strong as $\mathsf{KPi}$ even though $\textrm{min}\{\alpha\mid L_\alpha\vDash F\}$ is just the least limit of admissibles.
If this is true for $F$ being "the recursively inaccessibles are unbounded", then the least $\alpha$ such that $L_\alpha\vDash\mathsf{KP}+F$ is the least recursively-hyper-inaccessible. In the notation Degrees of Reflection $C(C(\Omega,C(\Omega,\Omega)),0)$ is assigned to this ordinal, so our proof-theoretic ordinal is conjectured to be $C(C(\Omega,C(C(\Omega,C(\Omega,\Omega)),0)),0)$.
About what this proof-theoretic ordinal would be using the ordinal collapsing function in the question, it depends on what map $\xi\mapsto I_\xi$ is, and also the function may not be adequate for representing the ordinal in the case that it has no way to construct the least hyper-inaccessible.