Maybe I misunderstand, but how can time derivatives survive if you send c to infinity? As far as I know, there is no non-static extension of the Poisson eqn. for the Newton potential.I don’t agree with this. Consider the Einstein-Infeld-Hoffman equations for approximating n-body motion in GR. These are based on the first order post Newtonian approximation. In the limit of c approaching infinite, they become exactly the Newtonian differential equation. Thus, there is a well defined Newtonian limit to a nonstatic GR situation.
But I'll check your link later, maybe that clarifies some things ;)