In this answer @PhillS linked to the paper Progress in Lunar Laser Ranging Tests of Relativistic Gravity which has given me a great start. While its main focus is on the Equivalence Principle (EP) they also mention that no local expansion was seen. The last sentence in the abstract is:
" The search for a time variation in the gravitational constant results in G/G ˙ = (4 ±9) ×10 −13 yr − 1 ; consequently there is no evidence for local ( ∼1 AU) scale expansion of the solar system."
Briefly, the data here is 34 years of laser ranging from Earth of a retroreflector array on the moon. While the distance between a given measurement site and the reflector on the moon can vary by as much as 50,000km (due mostly to the Moon's elliptical orbit and the size of the earth) these can be and have been painstakingly modeled. The amazing result is that in 34 years the residual scatter is only about 2cm!!
What I'm looking for is a midrange answer - not (exclusively) high level cosmology, but more than just balloon and raisin cake analogies. Something that will help towards understanding the relationship between the two.
Question: How is a result of no time varying $G$ related to a measurement of no local expansion?
Bonus mini-question: If I understand correctly and the null measurement of expansion is from the earth-moon ranging data, why does the sentence say "local (~1 AU) scale" when the earth-moon distance is only 0.0027 AU?
Lunar Laser Ranging images from here