We can always set up a coordinate system in an arbitrary manner[for example we may think in terms of spherical or rectangular systems as three dimensional time-slices]. Then we can find out metrics that match against the physical aspects of the problem[this should include gravity and perhaps other factors according to the nature of the problem].All this is over-complicated and missing key points. Coordinates have *no* meaning by themselves. They can have meaning only in conjunction with a metric expressed in them (note that the metric itself can be defined without reference to coordinates). In particular, there is no conceivable meaning to talking about different metrics on the same coordinates. If the actual physical situation is unchanged, what you mean is you've changed coordinates producing a changed expression of the metric. If the physical situation is different, then you have different coordinates *and* different metric - you just can't attach meaning to the coordinates in the abstract, separate from the metric.
The coordinate system is of course arbitrary----it does not have to have a definite physical meaning.But once we use the physical aspects of the problem to impose the metric coefficients on them,the whole thing becomes meaningful.
This in no way serves as any impediment to my suggestions/thought experiments.