If "meaningful" means measurable then seeng a half-percent change in the period of a tiny double-asteroid at a few AU is pretty close to as big as possible.
@PearsonArtPhotos question Using DART to measure G came to mind when I came across this "classic" question, so let's connect the dots.
From Double Asteroid Redirection Test (DART) Mission
DART will be the first demonstration of the kinetic impact technique to change the motion of an asteroid in space. The DART mission is in Phase B, led by JHU/APL and managed by the Planetary Missions Program Office at Marshall Space Flight Center for NASA’s Planetary Defense Coordination Office.
DART is a planetary defense-driven test of one of the technologies for preventing the Earth impact of a hazardous asteroid: the kinetic impactor. DART’s primary objective is to demonstrate a kinetic impact on a small asteroid. The binary near-Earth asteroid (65803) Didymos is the target for DART. While Didymos’ primary body is approximately 800 meters across, its secondary body (or “moonlet”) has a 150-meter size, which is more typical of the size of asteroids that could pose a more common hazard to Earth.
The DART spacecraft will achieve the kinetic impact by deliberately crashing itself into the moonlet at a speed of approximately 6 km/s, with the aid of an onboard camera and sophisticated autonomous navigation software. The collision will change the speed of the moonlet in its orbit around the main body by a fraction of one percent, enough to be measured using telescopes on Earth.
Wikipedia's Double Asteroid Redirection Test says that the launch mass is 500 kg, and the Lunar and Planetary Science XLVIII (2017) paper The Double Asteroid Redirection Test (DART) Element of the Asteroid Impact and Deflection Assessment (AIDA) Mission gives an impact mass of ~490 kg.
With a system mass $M = m_1+m_2$ of 5.28E+11 kg, a separation $R$ of 1180 meters, and the gravitational constant $G$ of 6.674E-11 m^3/kg s^2, the orbital period from (from here):
$$ T^2 = \frac{4 \pi^2 R^3} {G(m1+m2)} $$
is about 42,900 seconds, and if the orbit were circular that corresponds to an orbital velocity of about 0.173 m/sec.
The momentum of the ~500 kg spacecraft at 6,000 m/s is 3E+06 kg m/s, that of the moonlet in the system's center of mass (assuming the moonlet has about 0.66 % of the system mass if you assume equal density) is about 6.01E+08 m/s, so the complete absorption of momentum could change the momentum of the moonlet by roughly half of one percent.
Schematic of the DART mission shows the impact on the moonlet of asteroid (65803) Didymos. Post-impact observations from Earth-based optical telescopes and planetary radar would, in turn, measure the change in the moonlet’s orbit about the parent body.
The near-Earth asteroid (185851) 2000 DP107 in many ways is an analog to Didymos. 2000 DP107 was the first binary asteroid ever imaged by radar. This animation is derived from planetary radar observations. In this example (2000 DP107), the primary and secondary are about 850 meters and 300 meters in diameter. They are separated by about 2.7 km. The primary rotates once every 2.77 hours while the tidally locked secondary orbits the primary about once every 42 hours.
