Divisors of sequence $n,P(n),P(P(n)),\ldots$

Assume $P(0) \ne 0$. Let $e(m)$ be the smallest positive number such that $P^{e(m)}(0) \equiv 0$ mod $m$ (if it exists). If $S$ is the finite set of primes we suppose are allowed to divide $P^k(n)$ with arbitrarily high exponent, we can choose $x$ large enough that for all $p \in S$, $e(p^x) \ge |S| + 1$. Now in any contiguous subsequence of length $|S| + 1$ we can find at least one value that is not divisible by $p^x$ for any $p \in S$ (by the pigeonhole principle), but the largest possible such value is $\prod_{p \in S}p^{x-1}$, and the sequence is increasing without bound: contradiction.

Unfortunately I don't know what to do about the $P(0) = 0$ case. We can first write $P(x) = x^a \cdot Q(x)$ where $Q(0) \ne 0$ but I can't figure out how to adapt the above argument.