You can use the pre-$\lambda$ ring identity $$ \big(\sum_n [\mathrm{Sym}^n(X)]t^n\big)\big( \sum_k [\wedge^k(X)](-t)^k\big) = 1.$$ So it is enough to observe that the generating series for the exterior powers is a polynomial, for any finite $G$-set $X$. Here $\wedge^k(X)$ is the $G$-set of $k$-element subsets of $X$.

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, from which it in particular follows that $\lambda^k[X]=0$ if $k>|X|$.