  This work is the third part of a program initiated in arXiv:2111.13258,
arXiv:2302.06571 aiming at the development of an intrinsic geometric
well-posedness theory for Hamilton-Jacobi equations related to controlled
gradient flow problems in metric spaces. In this paper, we finish our analysis
in the context of Wasserstein gradient flows with underlying energy functional
satisfying McCann's condition. More prescisely, we establish that the value
function for a linearly controlled gradient flow problem whose running cost is
quadratic in the control variable and just continuous in the state variable
yields a viscosity solution to the Hamilton-Jacobi equation in terms of two
operators introduced in our former works, acting as rigorous upper and lower
bounds for the formal Hamiltonian at hand. The definition of these operators is
directly inspired by the Evolutional Variational Inequality formulation of
gradient flows (EVI): one of the main innovations of this work is to introduce
a controlled version of EVI, which turns out to be crucial in establishing
regularity properties, energy and metric bounds along optimzing sequences in
the controlled gradient flow problem that defines the candidate solution.
