@sweetser You are hoping to be able to derive a line element $ds^2 = \exp(-2w) dt^2 - \exp(2w)(dx^2 + dy^2 + dz^2)$ where $w = GM/R$. Correct? And you want to interpret $x=y=z=0$ as a point some constant distance away from a spherical mass. Correct? If you don't like the term "exponential metric", then we can call it something else. However that would mean there is no event horizon about a collapsed mass, no matter how dense.

So recent work on imaging a black hole directly could provide experimental evidence for or against your proposed line element since you disagree a lot with GR in the strong field.

This is a way to test your idea regardless of whether, or how, you end up being able to derive it from some theory you are working on. Its the closest we can get to getting at your base idea directly, since no matter what you change, you keep working to get this line element out as a prediction.