It is easey to see that the boundary conditions do nothing for you. Suppose one solution consistent with them. Do any of uncountably infinite coordinate transformations, you are still consistent with them *and* with your set up.A set of differential equations should have a unique solution set corresponding to a given set of boundary conditions. We may try out different techniques--but the aim is to find a solution set that fits into the boundary conditions.If we can do this--the job is done.We can get the correct solution from a set of infinite solutions.

You really need to let go of the idea of coordinate grid having any meaning (separate from a metric; or unless defined with a fixed operational definition). If you consult books on GR, you will find 100% unanimity that coordinates by themselves are meaningless. More, that points in spacetime have no meaning; only material objects and measurements have meaning.