**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. Let $W$ be the atomless part of $V$, the sets
that have no atoms in their transitive closures. It follows that
$W$ is a model of the usual ZFC, without atoms. Since $W$ is
definable, it follows that $j\upharpoonright W$ is an elementary
embedding from $W$ to $W$. I claim that it is nontrivial, which
will contradict Kunen's theorem.

So suppose $j$ is trivial on $W$, and in particular, $j$ fixes
every ordinal.

For any set of atoms $A$, define the rank hierarchy over $A$ by
$V_0(A)=A$, $V_{\alpha+1}(A)=P(V_\alpha(A)$ and
$V_\lambda(A)=\bigcup_{\alpha<\lambda}V_\alpha(A)$ at limits.

Since $j(A)$ is a set of atoms, but can contain only the atoms in
$A$, it follows that $j(A)=A$. Thus, since $j$ also fixes ordinals,
it follows that $j(V_\alpha(A))=V_\alpha(A)$. (This observation seems to address the concern you mentioned in your question.)

An inductive argument now shows that $j$ is the identity on the
elements of every $V_\alpha(A)$. If this is true at $\alpha$, then
it is true for the elements of $V_{\alpha+1}(A)$, and the statement
carries trivially through limits.

This implies $j(u)=u$ for every set $u$, since every $u$ is in
$V_\alpha(A)$ for the set $A$ of atoms appearing in its transitive
closure. 

So we've established that $j$ must move ordinals, and so $j\upharpoonright W:W\to W$ is a nontrivial elementary embedding of $W$. By restricting to $V_{\lambda+2}^W$, where $\lambda$ is the supremum of the critial sequence, we get a set embedding $j\upharpoonright V_{\lambda+2}:V_{\lambda+2}\to V_{\lambda+2}$, which is an element of $W$, and this contradicts the Kunen inconsistency in $W$.\Box$