Here's a rough answer that I'll clean up after seminar. Corollary 4.8 in Usuba's "A note on Lowenheim-Skolem cardinals" states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.
It is reasonable to doubt the consistency of the theory NBG + there is a Reinhardt cardinal + there is a proper class of Lowenheim-Skolem cardinals. Assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper "The HOD Dichotomy."