Take a look at the linked derivation by Eric Lord. According to my memory, the solution one normally gets differs from the weak gravity limit of the Schwarzschild solution in Schwarzschild coordinates by a coordinate transformation. Try to compare with the Schwarzschild solution in isotropic coordinates.

Method Poisson's equation from GR

The Schwarzschild metric is the exact, spherically symmetric and static, solution of Einstein's equations, in vacuum, where the only parameter is the mass of the object-so the energy-momentum tensor of matter vanishes.  You won't get a singularity-and, therefore, a horizon-with linear equations. The horizon is a *global* property of spacetime (as separating the observers that will not reach the singularity in finite proper time from those that will). 

Cf. https://arxiv.org/abs/gr-qc/9712019, p.164 and following. 

One can, also, show-but it's harder-that it's stable under certain perturbations:  http://www.ctc.cam.ac.uk/activities/adsgrav2014/Slides/Slides_Holzegel.pdf

and https://math.berkeley.edu/~phintz/kdsstab.pdf

If you do want to study perturbations about a Schwarzschild metric, just write g = g_S + h and keep the linear terms in h. 

People have written theses on the subject: http://uir.unisa.ac.za/bitstream/handle/10500/1667/dissertation.pdf?sequence=1

spacelike slices of the usual (3+1)-dimensional Schwarzschild metric (which might as well represent a spherically symmetric static black hole in vacuum) are themselves mere isolated objects made up of ordinary matter bound to be the classical vacuum solution of General Relativity. There is likelihood of manipulating them in a similar manner to the solution of the Newtonian differential equations. We may also abstractly justify the form of the metric which is used to then set up the Einstein field equations that possess the metric in Eddington-Finkelstein coordinates as their solution, and then directly solve these EFE to give the EF coordinates. But very unfortunate fact is that majority of investigators do not know they can facilely go up in dimension with the extension of their manifoldic tangents .