Let $F$ be a finite field. Let $F[X]$ and $F[[X]]$ denote the ring of polynomials and power series over $F$, respectively. I'm trying to show a statement like the following:

Let $g\in F[[X]]$. If there exists a set $C\subseteq F[X]$ of polynomials (with no constant term) of degree at most $k$ such that for all $c\in C$, $g(c)$ -- $g$ composed with $c$ -- is a degree $kd$ polynomial (for some fixed $d$), and $|C|$ is "large", then $g$ must actually be a polynomial.

I'm trying to beat the bound that one might be able to get via Schwartz-Zippel (it's not clear to me that one can even get a Schwartz-Zippel-type bound), where $|C| > kd |F|^{2k-1}$ (where $kd \ll |F|$).

What bounds on $|C|$ can we get?

Thank you, Henry