After failing to find any evidence that the notions I asked for have been previously defined, I chose to write things down. The resulting paper is available: A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces. Wasserstein spaces were my initial target, while (generalized) Hilbert cubes are handy reference spaces.
By the way, I should stress that using the family of functions $(x\mapsto \exp(-\lambda/x))_\lambda$ suggested in the question is a bad idea: the resulting analogue to Hausdorff dimension is not bi-Lipschitz invariant. One has to use cruder families like $(x\mapsto \exp(-x^{-s}))_s$.
Is it good policy to accept my own answer so that the question is not left open?
