I think the following is a counterexample to your specific question. Let AH be the set of those $x$ such that (1) each element of $TC\{x\}$ has cardinality at most $\aleph_\omega$ and (2) all but finitely many elements of $TC\{x\}$have cardinality strictly smaller than $\aleph_\omega$. (By $TC\{x\}$, I mean the transitive closure of the singleton, so it contains $x$, all its members, all their members, etc.) If $y\in x$ then $TC\{y\}$ is a subset of $TC\{x\}$, so AH is transitive. AH contains $\omega$ and is easily seen to be closed under pairing, union, and (binary) Cartesian product. Furthermore, it satisfies not only $\Sigma_0$-comprehension but full comprehension, because if $y$ is a subset of $x$ then $TC\{y\}$ is a subset of $TC\{x\}\cup\{y\}$. So AH is amenable.

For each natural number $n$, let $A_n=\{\{n\}\times\aleph_k:k\in\omega\}$, and notice that AH contains not only each of the $A_n$'s but also the set $X$ consisting of all the $A_n$'s. The union of any particular $A_n$ is $\{n\}\times\aleph_\omega$, which is in AH and has cardinality $\aleph_\omega$, but the set of all these unions is not in AH because it has these infinitely many elements of size $\aleph_\omega$. Summary: $X$ is in AH but $\{\bigcup z:z\in X\}$ is not.

Comment: If one modifies the definition of AH by requiring all elements of $TC\{x\}$ to have size strictly below $\aleph_\omega$, one gets the standard example of a model of all the ZFC axioms except the axiom of union. By allowing, in the definition of AH, finitely many exceptions of size $\aleph_\omega$ one revives the axiom of union and in particular one lets each of the sets $\bigcup A_n$ into AH (but just barely) but not the collection of all of them.