Tim's argument is correct, and here's a different way to see this. To say that $\Sigma^{\infty} BS$ is a finite $p$-group has trivial rationalization and is $p$-local is the same as to observe that it is $p$-torsion, i.e. has $p$-torsion homotopy groups.
To see without involving the Serre spectral sequence, observe that both the rational and $\mathbb{Z}/l$ cohomology (for $l \neq p$) of $\Sigma^{\infty} BS$ vanishes by Maschke's theorem. For a connective spectrum, this forces it to be $p$-torsion, by the Hurewicz theorem.
We also have that $F(T, U)$ is p-complete for any $p$-torsion $T$ and arbitrary $U$.
To see this, observe that the subcategory of $p$-torsion spectra is generated under colimits and desuspensions by $M(p)$ (this is the same as saying that any such non-zero spectrum admits a non-zero map from $M(p)$, which is clear). Since $F(-, U)$ takes colimits to limits, and $p$-complete spectra are closed under limits, we then just need to know $F(S^{0}/p, U)$ is $p$-complete, but this is the same as $\Sigma^{-1} U \otimes S^{0}/p$, so we're done.