I agree with your comment on uniqueness. But it's not so clear to me how the Bamler-Brendle paper that Otis linked to answers the question. It seems to me like it's answering a different question.

Also I'm curious why you say that $m$ is usually negative? My impression was that the loss of derivatives is typically somewhat small, at least small enough that you gain some regularity in passing from the RHS to the solution. Is this wrong?

Like I asked Tobias in my second comment above, isn't there some technical difficulty about which function spaces one identifies for the inverse map to be smooth? In Hamilton's paper, it seems that the only conclusion is that the map $C^\infty\to C^\infty$ is smooth and tame, and it seems like the Hamilton-Nash proof does not (and possibly is incapable of?) finding any kind of optimal $m$ for $C^{k,\alpha}\to C^{k+m,\alpha}$ to be smooth. But maybe I don't understand it properly.

@DeaneYang Is it so simple even in the elliptic case? In well-known cases like e.g. the Calabi-Yau complex Monge-Ampère equation, even once one has a solution map $C^{k,\alpha}\to C^{k+2,\alpha}$ and the PDE is known to be uniformly elliptic at the solutions, it doesn't seem totally obvious that the map $C^{k,\alpha}\to C^{k+2,\alpha}$ is smooth. (Is it smooth? It seems like it could follow from estimates for the linearized equation, but it isn't obvious to me how. Is it analytic?)

Aren't there still two differences? 1) Hamilton's conclusions assert tameness, but nothing about the degree of tameness; 2) the conclusion on smoothness is as a map $C^\infty\to C^\infty$, which doesn't (?) automatically extend to smoothness as a map $C^{k,\alpha}\to C^{k+m,\alpha}$

I'm not very familiar with Hamilton's paper, but doesn't that only cover the case of linear $P$?