Can we redrive Schwarschild metric from Kepler's law without assuming General Relativity?
Somewhat surprisingly there is a sort of derivation of the Schwarzschild metric from Kepler's third law. Rather than give the (lengthy!) details here let me refer you to this article that works through the calculation.
But this is not really a derivation because it assumes that the $r$ in Kepler's law is the same as the Schwarzschild $r$ coordinate, and this is not the case. It also assumes that the $t$ in Kepler's law is the same as the proper time, and again this is not true.
The result is a coincidence due to the way the Schwarzschild $r$ coordinate has been chosen There are several such coincidences. For example we have the coincidence that the event horizon position is the value of $r$ for which the Newtonian escape velocity is equal to the speed of light, and again this is due to the way the Schwarzschild $r$ coordinate has been chosen.