Perhaps you can go even further and try to make it so that any two points have pairwise correct distances, like in the Azimuthal equidistant projection?

https://en.wikipedia.org/wiki/Azimuthal_equidistant_projecti...

Perhaps there are metric functions where this is not possible (a quick proof eludes me...[1]), but approximations might be possible anyway.

[1] Edit: actually it's quite simple. Any non-isotropic metric cannot have an Euclidean equidistant projection, simply take a pair of points lying in rays (1,0) and (0,1) in an L^p metric, p not 2, as a counterexample.