Working in $\textsf{ZF} + \text {there is a strongly inaccessible cardinal}$.

Let $\kappa$ be the first strongly inaccessible cardinal, and let $\lvert V_\kappa\rvert= \kappa$, then $(V_{\kappa+1}, \in)$ would be a model of $\textsf{MK}$.

Now, is it consistent to add that $V_{\kappa+1}$ is non-well-orderable?

If yes, then what's the benefit of having $V_{\kappa+1}$ well-orderable, on the theory $\textsf{MK}$? I mean what additional axioms in the language of $\textsf{MK}$ would this confer?