Gravity from the viewpoint of theory of sources

I examine the $G^2$-approximation of Schwarzschild solution from the viewpoint of theory of sources. The method suggests the following definition of the privileged coordinate system: it is a system in which in each approximation the gauge degrees of freedom are put to zero, i.e. the metric is formed solely by sources. I calculate the metric in this system. In $G^2$-approximation the exterior metric has the term which is of the form of a gauge function. Considering it as such I have the agreement with Schwarzschild metric. But I cannot consider it as a gauge function because it is generated by sources. It should be observable. It is proportional to the radius of matter ball and seems violate the Birckhoff theorem.