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What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from The spectrum of elementrary embeddings j : V → V by Paul Corazza?

BTEE uses the language of $(V,∈,j)$ and asserts:
ZFC (without separation and replacement for formulas using $j$)
(schema) $j$ a is nontrivial elementary embedding of $V$ into $V$ (for formulas without $j$)
critical point: the least ordinal moved by $j$ exists.

Without critical point, the theory would be conservative over ZFC. With full replacement (and given the axiom of choice), the theory would be inconsistent; and there is a hierarchy between the two extremes.

BTEE suffices for a basic theory of $j$. For every natural number $n$, BTEE proves that $κ = \mathrm{crit}(j)$ is $n$-ineffable and totally indescribable (see the linked paper). However, the only upper bound on the strength of BTEE in the paper is $ω$-Erdős.

This question is in Q/A format.

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BTEE is conservative over the stationary reflection principle, i.e. ZFC + (schema) {there is $n$-subtle cardinal}$_{n∈\mathbb{N}}$ . (Note that using $n$-ineffable in the schema is equivalent.) As noted above, for every $n$, BTEE implies existence of $n$-subtle cardinals.

For the converse, given a proposition $A$, consistency of ZFC + $A$ + (schema) {there is $n$-subtle cardinal}$_{n∈\mathbb{N}}$ implies existence of a model $M$ of ZFC + $A$ + $n$-subtle cardinal for a nonstandard number $n$, plus existence of $λ$ with $V_λ^M≺M$. Fix such an $M$ and an $M$-well-ordering $<^*$ of $V_λ^M$, and (in $M$) use an $n$-subtle cardinal $<λ$ to produce an $n$-tuple $(κ_1,...,κ_n)$ of good indiscernibles for $(V_λ^M, ∈^M, <^*)$. ("good" is a technical term here and is why we need the $n$-subtle cardinal.) Let $M'$ be the $(V_λ^M, ∈^M, <^*)$ Skolem hull of the first $ω$ of these indiscernibles (as computed in $V$ so all Gödel numbers are standard). $(M',∈^M,j)$ satisfies BTEE + $A$ where $j(κ_i)=κ_{i+1}$ and $j$ is extended to all sets in $M'$ using the Skolem hull.

Adding induction (and especially transfinite induction) would significantly increase the strength. For the strength of existence of a well-founded model of BTEE, see this question.

$n$-huge cardinals

Nonstandard models can also be used to show that WA0 (bounded quantifier Wholeness Axiom, which consists of BTEE and separation for bounded quantifier $j$-formulas) is conservative over ZFC + (schema) {there is $n$-huge $κ$ with $V_κ≺_{Σ_n}V$}$_{n∈\mathbb{N}}$ . (In turn, this is $Σ_2^V$ conservative over ZFC + (schema) {there is $n$-huge $κ$}$_{n∈\mathbb{N}}$ .)

One direction is proved in the linked paper. For the converse, given a proposition $A$, consistency of ZFC + $A$ + (schema) {there is $n$-huge $κ$ with $V_κ≺_{Σ_n}V$}$_{n∈\mathbb{N}}$ implies existence of a model $M$ of ZFC + $A$ + $n$-huge cardinal $κ$ for a nonstandard number $n$, and with $V_κ^M≺M$. Starting with such $M$ and an $n$-huge embedding $j$ in $M$ with $κ = \mathrm{crit}(j)$, let $M' = \{x∈M: ∃m<ω \,\, x ∈^M j^m(V_κ^M)\}$. (Note that we use $ω$ rather than $ω^M$.) $(M', ∈^M, j)$ satisfies WA0 + $A$.

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