I don't know about the complexity, but I can at least point out that finding an isomorphism from $k[y]/(Q)$ to $k[x]/(P)$ is the same as finding a Tschirnhaus transformation from $P$ to $Q$. Furthermore, this can be algorithmically tested, at least in theory, by embedding them in the decomposition field of $PQ$, computing the Galois group $G$ of the latter, and checking whether the associated subgroups are conjugate in $G$: so at least it's decidable.
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