I think there's a misunderstanding here: there is one method for building a model in which choice breaks, and the point is that to get a model in which choice fails and some large cardinal axiom $(*)$ holds, we start with a model in which $\kappa$ has $(*)$, and then cause a failure of AC sufficiently far above $\kappa$ that $\kappa$ still satisfies $(*)$ in the resulting model.
In this example, if there is an elementary embedding $j$ of $L(V_{\lambda+1})$ into itself with critical point below $\lambda$, then $j$ is also an elementary embedding of $L(W_{\lambda+1})$ into itself with critical point below $\lambda$ as long as $W$ is any forcing extension of $V$ such that $W_{\lambda+1}=V_{\lambda+1}$. So if we start with a $V\models I_0$, and add a failure of choice by taking a symmetric submodel of an extension of $V$ by a forcing which is sufficiently closed (so doesn't alter $V_{\lambda+1}$), the resulting model will still satisfy $I_0$.
EDITED FOR CLARITY: Note that all we're using is that the $I_0$ axiom depends only on an initial segment of the universe; so breaking choice above that segment won't affect things. This isn't entirely obvious, since unlike e.g. inaccessibility, the $I_0$ axiom seems to refer to a proper-class sized object - namely, $L(V_{\lambda+1})$ and the embedding $j$. However, the statement "$x\in L(y)$" is absolute, and so things are no more complicated (from a killing-choice-while-preserving-the-axiom point of view) than the case of $I_1$, which is explicitly about the set $V_\lambda$.
In principle, we could imagine variations on the $I_0$ axiom of the following form: "There is an elementary embedding of $M(V_{\lambda+1})$ into itself with critical point below $\lambda$," where "$M(-)$" is shorthand for "the class of $x$ such that $\varphi(x, -)$" for some $\varphi$. So long as $M(-)$ is always guaranteed to be an inner model, this could be interesting, and if (the formula $\varphi$ associated to) $M$ were sufficiently non-absolute, then there wouldn't be an obvious way to kill choice while preserving "$I_0(M)$". However, I'm unaware of anything interesting along these lines.
