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

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

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 $\sf MK$? I mean what additional axioms in the language of $\sf MK$ would this confer?