It seems to me that every nonempty computably enumerable set will be the range of a primitive recursive function that is very low in the Grzegorczyk hierarchy, and it seems that even $\cal {E}^1$ suffices. The reason is that if $A$ is a nonempty c.e. set, then it is the domain of some computable function $\varphi_e$, with Turing machine program $e$. Let $k_0$ be the smallest element of $A$. Let $p$ be the primitive recursive function defined so that $p(n)=k$ if $n$ is a Gödel code of the entire computation sequence of program $e$ on input $k$, and this computation sequence successfully attains the halt state, and otherwise $p(n)=k_0$ if $n$ is not such a code. The graph of this function is $\Delta_0$-definable and so $p$ really is primitive recursive. Furthermore, $p$ is not at all a fast growing function, since $p(n)$ is much less than $\max\{n,k_0\}$, and so $p$ arises in very low levels of the Grzegorczyk hierarchy. Specifically, $p(n)$ is definable by recursions that do not need to consult numbers larger than $\max\{n,k_0\}$. So it seems that $p$ is in level $\cal{E}^1$, at the bottom of the hierarchy. Meanwhile, the range of $p$ is precisely $A$, as desired.

(Note that we may slow the growth of $p$ down further, as much as desired, by requiring $n$ to code much more information, before giving the comparatively tiny output $k$.)