You were probably mainly interested in the case of $a\le n$; the other case of $a>n$ seems to be false. Here's a quick counterexample for $a>n$, letting $n=1$ and $a=2$. Notice then that $h_2 = (h_1)^2$.
If there were a factorization with $a\le n$, it would need to involve nonsymmetric polynomials. But since $S$ is a unique factorization domain, this would mean for any nonsymmetric factor, we would also need all the elements of its orbit as factors. While I was editing this, Will Sawin came up with an elegant proof, whereas I was just grinding through cases, so I will omit those cases.
Since Will mentions still needing the base case of $n=a=3$, notice that we would need a nonsymmetric factor, which means a nonsymmetric factor with either (a) terms of the form $x_ix_j^2$, with nontrivial orbit, yielding too high a degree, or (b) it has terms of the form $x_i$ with some included and some omitted, contradicting the product including all $x_i^3$ terms, or (c) it has some terms $x_ix_j$ with $i\ne j$ and omits others, again necessitating other factors in the orbit, giving too high a degree for the product, or (d) it has some terms $x_i^2$ and not others, giving the same contradiction.