Theorem. The Kunen inconsistency works over ZFC with atoms. That is, in this theory, there is no non-identity elementary embedding $j:V\to V$ that fixes every atom.
Proof. Suppose that $j:V\to V$ is an elementary embedding fixing every atom. If $j$ is not the identity embedding, then I claim that $j$ must move an ordinal. To see this, let $u$ be any $\in$-minimal element that is moved by $j$. Let us well-order $u$ in some order type $\beta$. Since ordinals are fixed by $j$, it follows that the $\alpha^{th}$ element of $u$ is carried by $j$ to the $j(\alpha)^{th}=\alpha^{th}$ element of $j(u)$, but since those elements are fixed (by minimality of $u$) and the total length of the well-order is fixed, it follows that $j(u)$ is simply $u$ itself, contradicting the choice of $u$. (In fact, by an argument on ranks, one can eliminate the use of choice in this part of the argument; but you still need it for the other part.)
So $j$ is not the identity on ordinals. We may now restrict $j$ to the pure part of the universe. Let $W$ be the class of sets having no atoms in their transitive closures. This is a model of ZFC, without atoms. Since this is definable, it follows that $j\upharpoonright W$ is a nontrivial elementary embedding from $W$ to $W$. This contradicts the usual Kunen inconsistency. For example, one gets that $j\upharpoonright V_{\lambda+2}^W$ is a set in $W$, where $\lambda$ is the supremum of the critical sequence, and this violates the usual ZFC version of the Kunen inconsistency in $W$. $\Box$