The hypothetical, non-existent crowdfunding site go-launch-me is so confident, they are offering a two-for-one special. For a limited time, if your project is funded they will foot the bill for a 2nd project of identical cost and complexity.

I've decided to propose a Gold Ball project, an $m=$ 1000 kg sphere with a tiny laser interferometer and small camera on one side (and an equal one on the other for symmetry) to be launched into space and put in a distant retrograde orbit around the Earth, or perhaps heliocentric at or moving towards Sun-Earth L4 or L5. Some place far from strong gravity gradients.

Per the site's terms, if funded, they will pay for a second identical ball, so I will have two optically polished gold balls in space.

I propose that they be put in a tight orbit around each other. I estimate that the radii of the balls is $r=0.23$ meters, if they are placed at $R=0.5$ meters center-to-center there will be a 4 cm average gap between them if they orbit their common center of mass, their orbital period should be:

$$ T^2 = \frac{4 \pi^2 R^3} {2GM} $$

or about 101 minutes.

Wikipedia gives the value of G as

$$6.67408(31) \times 10^{-11} m^3 kg^{-1} s^{-2}$$

and the 31 is the uncertainty in the last two decimal places, so its roughly 50 ppm. The article goes on to report two very recent measurements at 12 ppm but they differ by almost three times that.

Their two laser interferometers are at different wavelengths, and so you can more easily measure the absolute surface-to-surface distance without the integer-fringe ambiguity that you might get from a single wavelength.

The tiny cameras in each sphere image the reflection of the Sun in the convex, polished gold surface of the other sphere, so precise angular information versus time is measured along with precise separation information. The orbits will be slightly elliptical (nothing is perfect) but that's pretty easy to account for.

Effects of solar photonic and wind pressure on their mutual 101 minute orbit cancel, since the spheres are identical.

Question: What could go wrong? Why might these gold balls not be a great way to measure G? Are there secondary forces or torques that spacecraft experience that could interfere with the measurement, or aspects of spacecraft design that would complicate the design of the golden balls beyond practicality?

To address @RussellBorogove's concern that this is too hard and should be in Physics SE, in the first experiment we'll measure the sum of both the "normal" $-\frac{GM}{r}$ potential plus any yet-undiscovered short-range gravitational effects.

note: hat tip to @rob for mentioning their answer in Physics SE which discusses a real-world consideration of such an experiment!