Thanks! This basically satisfies my curiosity, so I'll mark it as resolved. One comment is that these references (and work cited-therein by Basu and Khatri (1969)) all assume the function is bijective. It isn't clear to me if this is necessary, and from a quick skim, the authors don't seem to remark on it.

My answer still works. X,Y,Z are pairwise independent in this construction.

I guess you might already know, but this is closely related to McKean's conjecture, a weak form of which speculates that the successive derivatives of $H(X_t)$ alternate in sign. See "Higher Order Derivatives in Costa’s Entropy Power Inequality" by Cheng and Geng, and references therein.

to add to above, this equivalence does not actually make your problem any easier. It just converts from an entropy problem to an equivalent functional one. Except in special cases (e.g., where $P$ is product of rho-correlated Gaussians or Rademachers), a characterization of the extremizers is likely non-explicit. However, if $P$ is a product measure, you can use tensorization of Brascamp--Lieb inequalities (and their entropic equivalents) to conclude that the extremizers will also have product structure. You should also be able to see this directly.