The question asks whether it is possible to mathematically prove the speed of propagation of gravity waves without linearization. 

The answer is No, as observed by Dirac, Eddington, and Einstein himself (see references below). The key point is that Einstein's mathematical expression for the so-called "energy components" $t^\alpha_\sigma$ of the gravitational field is not tensorial. The non-tensoriality of the energy components implies the energy field cannot be localized and has no observer-independant components. 

P.A.M. Dirac said ``in general, gravitational energy cannot be localized. The best we can do is use a pseudotensor...which gives us approximate information about gravitational energy, which in some special cases can be accurate." (See Dirac's book "General Theory of Relativity"). 

A.S. Eddington similarly concluded the nonpossibility, writing: ``If coordinates are chosen so as to satisfy a certain condition which has no very clear geometrical importance, the speed [of gravity waves] is that of light; if the coordinates are slightly different the speed is altogether different from that of light. The result stands or falls by the choice of coordinates, ...". (See Eddington, The Mathematical Theory of Relativity, S 57). 

The key point -- as realized by Einstein, Eddington, Dirac, Hoyle, Abrams, even Crothers -- is that Einstein's so-called "gravitational energy tensor" is *not* a tensor at all! To quote Einstein: "The quantities $t^\alpha_\sigma$ we call the 'energy components' of the gravitational field,..., it is to be noted that $t^\alpha_\sigma$ is not a tensor". (See Einstein's ``The Foundation of the General Relativity, 1916, S.15). 

Einstein noted well that $t$ is not a tensor, but is invariant under *linear unimodular* change of coordinates. This is elaborated in many excellent articles by E.Norton (see his articles on General Covariance and Einstein's Point-Coincidence Argument, and the long documented struggles which Einstein had in developing satisfactory covariant equations. 

A further difficult is that Einstein apparently discovers the conservation of gravitational energy by evaluating the coordinate divergence(!) of $t$ and finding $\partial t^\alpha_\sigma / \partial x_\alpha=0$. He says "This equation expresses the law of conservation of momentum and of energy for the gravitational field." (Ibid) However, the vanishing of a coordinate divergence of a nontensor object is not a covariant object except (in this case) for observers who share the same volume form, i.e. unimodular linear change of coordinates change of coordinates. 

Dirac says, "Let us consider the energy of these waves. Oweing to the pseudo-tensor not being a real tensor, we do not get, in general, a clear result independant of the coordinate system. But there is one special case in which we do get a clear result; namely when the waves are all moving in the same direction", (Ibid). 

Any person interested in further critical analysis is well recommended to see Section 8 of Crothers' article https://vixra.org/abs/1804.0399 and also https://vixra.org/abs/1103.0051 . (This author's own critical review of the above articles finds them very informative, and with very clear treatment of differential geometry.