The question asks whether it is possible to mathematically prove the speed of propagation of gravity waves without linearization.
I humbly submit that the answer is No, and that it is proven by the following fact: that Einstein's mathematical expression for the so-called "energy components" $t^\alpha_\sigma$ of the gravitational field are not tensorial (references given below).
The non-tensoriality of the energy components implies that the energy field cannot be localized (and has no observer-independant components).
So the speed of propagation of the GW cannot be proven without an arbitrary choice of coordinates (and in that choice you have your choice of linearization).
Formally speaking, the statement "that energy components of the gravitational field satisfies wave equations" is not tensorial ("covariant") and therefore not a well formed statement in the category of Einstein's GR.
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, S 57).
The key point -- as realized by Einstein, Eddington, Dirac, 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 $\frac{\partial t^\alpha_\sigma}{\partial x_\alpha}=0$. He says "This equation expresses the law of conservation of momentum and of energy for the gravitational field." (Ibid) The vanishing of a coordinate divergence of a nontensor object is not a covariant object except for observers who share the same volume form, i.e. except under linear unimodular 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).
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.