Hi James,
This is probably going to hook me into a longer discussion than I am prepared to carry on, so I can't promise to stay with it. However, the question of a universal gravitational constant derived from Newton's gravity seems a good place to start.
You note that Newton's two-body gravity formula (eqn 36) incorporates two terms for mass, while his energy formula (eqn 37) has only one. The reason is that gravity describes a curvilinear relation between spatially separated points of mass changing in time, and force describes a point of mass changing linearly in time.
Taking the moon in relation to Earth, Newton found that the body is accelerating sufficiently in a curvilinear path to avoid colliding with Earth, yet not so fast as to escape orbit. The energy of acceleration, therefore, is countered by the negatively valued presence of Earth's mass; the inertia of Earth's rest mass, IOW, is sufficiently large to keep the inertia of the moon's accelerated mass falling in a curved path, rather than continuing in a straight line. Galileo had earlier found that objects in a gravity field fall at the same rate whether in a straight line or a curve, so Einstein recognized that it makes no difference calculationally whether the moon is accelerating toward Earth or Earth accelerating toward the moon -- gravity and acceleration are equivalent.
You speak of an "awkwardness" between eqns 36 and 37, and propose to convert Newton's linear f = ma, which is easy to solve, to a differential equation (eqn 38) which is not only diffcult to solve but to which you want to make Newton's also easy to solve 2-body gravitation relation equivalent (equation 39). I have to ask -- why? It appears to me you are trying to solve a problem that doesn't exist.
Relativity already describes mass-energy relations in terms of momentum. The rest energy equation E = mc^2 is quoted so often that one forgets its derivation: E^2 = m^2c^4 (pc)^2 where p is momentum. The unreduced equation tells us that a relativistic particle of zero momentum contains negative mass. (That's what allows me to speak of "negatively valued" mass in relation to Newton's theory above as a convenient fiction, a relativistic convention.)
I appreciate that you are motivated by the idea that mass is not a "given." That mass is made of space and time alone. That is what continuous field theories already tell us, though -- energy densities that vary from point to point of spacetime are massless when considered, as you put it, from the POV of a remote observer.
You're right that a variable speed of light eliminates the gravitational field -- but then, one also loses the measurement standard by which we determine that energy varies point to point. So your result (eqns 61/62) ends up saying that gravity does not vary in time, because the universal acceleration of gravity is constant (though in relation to what, since acceleration describes the rate of change of the rate of change?) and the universe is therefore static. I don't know how you reconcile that with an expanding universe, which is the same problem Einstein had when he introduced the cosmological constant. We believe today that the cosmological constant does not significantly differ from zero.
As near as I can understand you, James, you have substituted a hypothetical variable speed of light for the variation of energy densities in the spacetime field. I can't see a problem with this, relativistically speaking, and I don't have to check the math to agree with the idea in principle -- yet what is gained? We were always free to see mass as "slow light." This only works in one direction, however -- (initial photon veolcity is always the speed of light, the speed at which photons are created) -- when you start treating nuclear particles (eqns 51 - 55) in terms of positive acceleration, you are contradicting your original proposal to convert Newtonian mechanics to quantum mechanics. You identify no limit nor mechanism by which photons accelerating negatively become mass that accelerates positively. If photon acceleration degenerates to zero (as it must in your model) how does that differ from the singularity of general relativity?
What am I missing?
Tom