Firstly, I'm going to assume you want $x'$ to be nonzero. Secondly, do you want $x$ and $xx'$ to also both be non-zero?
If so, then there is a silly counterexample -- take $n=1$, and take $x$ to be $id - t$ where $t$ is the permutation $1\to 2\to \dots \to m\to 1$. Then $x$ lies in the augmentation ideal of $F[S_{m+n}]$ and so if $xx'$ lies in $\theta^*(F[S_1]) = F$ then it would have to be zero.
I strongly suspect that one could play similar tricks with higher values of $n$.