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

The answer is apparently no, and even more, the speed of propagation cannot be proven without an *arbitrary* choice of coordinates. So the statement is not even a proper statement in the category of GR, i.e. is not tensorial.

P.A.M. Dirac apparently believed it was *not* possible, and that ``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, section 57). 

The key point -- as realized by Einstein, Eddington, Dirac, and 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).

However Einstein is aware that $t^\alpha_\sigma$ acts like a tensor when restricted to unimodular linear change of coordinates. Crothers' key criticism is that Einstein then takes coordinate divergence of $t$, and not a tensor divergence (since $t$ is not a tensor). 

As 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). 

See Section 8 of Crothers article https://vixra.org/abs/1804.0399 for a detailed analysis, and also https://vixra.org/abs/1103.0051

(N.B. Regardless of your personal opinions as to the infallibility of great *men* of science, it appears that Mr. Crothers' article is *mathematically* sound. And for open minded persons honestly interested in the first principles of GR, his articles are also extremely informative.)