This is not a complete answer, but a comment that's gotten too long plus a hint.

My hunch is that it is unlikely that there is a general, simple connection. My reasoning is the following:

Let $X$ be a smooth manifold. The Fisher-Rao metric is invariant under the action of diffeomorphisms by pushforward, while the Fisher information functional $I$ is not; even under scalings on $X=\mathbb{R}^n$, the squared norm of the gradient will pick up a non-trivial scaling factor.

But maybe there is hope to find an interesting relation when the statistical manifold is closed under isometries, as you hint at with your translation example.

This is not a complete answer, but a comment that's gotten too long plus a hint.

My hunch is that it is unlikely that there is a general, simple connection. My reasoning is the following:

Let $X$ be a smooth manifold. The Fisher-Rao metric is invariant under the action of diffeomorphisms by pushforward, while the Fisher information functional $I$ is not; even under scalings on $X=\mathbb{R}^n$, the squared norm of the gradient will pick up a non-trivial scaling factor.

But maybe there is hope to find an interesting relation when the statistical manifold is closed under isometries, as you hint at with your translation example.

ADDENDUM: I found the following paper: B. Khesin, G. Misiolek, and K. Modin, “Geometric hydrodynamics via Madelung transform,” Proc. Natl. Acad. Sci. U.S.A., vol. 115, no. 24, pp. 6165–6170, Jun. 2018.

In Proposition 13 there, the Fisher metric and Fisher functional interact to produce some sort of wave equation that seems to be of interest. I'm afraid I cannot say much more. Maybe someone else will read this and be able to expand on this direction.