I have edited my answer above to answer your comment.

Taking into account the aforementioned multidimensional difficulty, it is indeed straightforward to prove (5) (which is the requested estimate) inductively.

@T.Amdeberhan I wrote some more details on the meaning of the multidimensional Faa de Bruno formula.

Hmm, I guess so. In fact, I hope that the formula $(\ast)$ can be used to prove this inductively, by just plugging your estimate for the $\Psi_r$, since the $\Omega_{r,n}$ have an explicit expression. I will return to that matter, but I feel that $(\ast)$ is following the piece of advice given in a previous comment by A. Kulikov.