This article provides non-trivial technical ingredients for the article "The quantitative hydrodynamic limit of the Kawasaki dynamics" by the same authors. In that work a quantitative version of the hydrodynamic limit is deduced using a refinement of the two-scale approach. In this work, we deduce the strict convexity of the coarse-grained Hamiltonian, a uniform logarithmic Sobolev inequality for the canonical ensemble and the convergence of the coarse-grained Hamiltonian to the macroscopic free energy of the system. We deduce those results following an approach developed by Grunewald, Otto, Villani and Westdickenberg. Because in our setting the associated coarse-graining operator is non-local, the arguments are much more subtle and need additional ingredients like the Brascamp-Lieb inequality and a multivariate local central-limit theorem.