Can we "invert" Density Functional Theory through sufficiently accurate experiment?

The famous Hohenberg-Kohn theorems say that there is a one-to-one mapping between the many-body Hamiltonian, $mathcal{H}$, of a solid and its ground-state electron density $rho(mathbf{r})$. As far as I understand, this also means that all the properties of the ground-state wavefunction are encoded in the electron density itself (though perhaps not in a simple way).

Density functional theory aims to solve for this ground-state electron density $rho(mathbf{r})$ through various simplifications and manipulations of $mathcal{H}$ to make the process computationally tractable.

I am interested in the reverse process, where an experimentalist comes up to me with their measured $rho(mathbf{r})$. In principle, a sufficiently accurate measurement of the electron density can be done with X-ray scattering (or electron microscopy) to obtain $rho(mathbf{r})$. Typically, such measurements of $rho(mathbf{r})$ are only used to get the positions of the atoms in a crystal, but the Hohenberg-Kohn theorems and DFT suggest you could do a lot more with $rho(mathbf{r})$.

So my question is: Given an experimentally determined $rho(mathbf{r})$ to arbitrary accuracy, what can we say about a material’s properties using "inverse" DFT?

As a followup, what experimental accuracy for $rho(mathbf{r})$ is needed to accurately determine those material properties?