= Solution
Realize $\xi^\lambda$ as the permutation character on ordered set partitions with row sizes $\lambda_1,\lambda_2,\ldots$. Under $S_m\times S_k$, an orbit is determined by the weak composition $\mu$ whose $i$th part counts elements of $\{m+1,\ldots,n\}$ in row $i$. Its stabilizer is the product of the Young subgroups for $\lambda-\mu$ and $\mu$. The orbit character is therefore the outer tensor product $\xi^{\lambda-\mu}\mathbin\#\xi^\mu$, and summing the orbits gives
$$
\left.\xi^\lambda\right\downarrow_{S_m\times S_k}
=\sum_{\mu\models k}\xi^{\lambda-\mu}\mathbin\#\xi^\mu,
$$
with impossible compositions contributing zero.
Insert this identity into the alternating definition of $\psi^\lambda$. Group the weak compositions by permutations of their parts and use <character straightening>; the alternating sum in the first tensor factor is $\psi^{\lambda-\mu}$, while the second factors combine once for each partition $\mu\vdash k$. Thus
$$
\left.\psi^\lambda\right\downarrow_{S_m\times S_k}
=\sum_{\mu\vdash k}\psi^{\lambda-\mu}\mathbin\#\xi^\mu.
$$
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