Re: Is there a value of G which satisfies the Newtonian problem of prehellion.

This is an interesting question, but I'm not sure exactly where it's coming from. You are correct that one of the great successes of general relativity is its prediction of the precession of the perihelion of Mercury. In plain language, this means that the closest approach to the sun of Mercury's elliptical orbit is drifting. The only discussions that I have found about a variable gravitational constant have it varying in time, which would not affect our observations of Mercury.

In Newtonian mechanics, we expect every closed orbit to be an ellipse - at least for two isolated bodies orbiting each other. The solar system is a more complicated environment, because in addition to the sun and Mercury, there are eight other planets and many, many small bodies. It turns out that the perturbations these other bodies induce on Mercury's orbit also cause a precession - in particular, Jupiter has a fairly large effect. But these effects can be calculated quite precisely, and they fall 43 arcseconds/century short of the observed precession. This discrepancy is exactly predicted by general relativity, and is what you would need to account for with a modified Newtonian theory.

Changing the value of G would not affect the two-body orbit in the way you need. It is the overall structure of Newton's equations of motion and the gravitational force that make bound orbits elliptical. The exact value of the constants doesn't affect the fact that two-body orbits are ellipses with no precession. It is probably possible to vary G with distance in such a way that you could reproduce the Mercury's precession when you add in the perturbations of the other planets. However, you would have to do so by hand, specifically to solve this problem. Then you'd have to match that theory to other observations (of gravitational lensing and the precession of the perihelion of Icarus, an asteroid, for example). Although I haven't seen it tried, my instinct is that varying G so as to solve the Mercury problem would not allow agreement with these other observations. General relativity, on the other hand, correctly describes all of these phenomena (and others).

Chaos theory has very little to do with this problem. That theory describes systems with "sensitive dependence on initial conditions," or systems in which a very slight change in the initial setup can have a drastic change on long-term consequences. While it is known that some solar system setups would be chaotic, the solar system is so old that we can safely say that the current configuration is quite stable. This means we do not need to worry about possible chaos when modeling this problem.