Any model of $ZF$+stationary proper class of $I3$ ordinals (whose consistency strength is at most $ZF$+$I2$) where $W_α$ is listing of $V_{κ}$ where $κ$ is $I3$ and $j_α$ the witness of $I3(V_κ)$ will be a model of your theory.

Injectivity and Cumulation are trivial, $ZF$+Elementarity is part of our assumption.

For Reflection, let $φ$ be any formula, by Levy's reflection principle there is a proper class club $C_φ$ that reflect $φ$ (note that Levy's reflection works even with a modified language).

Because there is a stationary set of $I3$ cardinals, there exists a $I3$ cardinal in $C_φ$, then this ordinal will witness Reflection of $φ$.