You can use the pre-$\lambda$ ring identity
$$ \big(\sum_n [\mathrm{Sym}^n(X)]t^n\big)\big( \sum_k \lambda^k[X](-t)^k\big) = 1.$$
So it is enough to show that the generating series for the exterior powers is a polynomial, for any finite $G$-set $X$. There is an explicit formula for $\lambda^k[X]$ in a paper of [Rökaeus][1], from which it in particular follows that $\lambda^k[X]=0$ if $k>|X|$.


  [1]: https://arxiv.org/abs/0708.1470