Such measurements have been done, using lasers reflecting off mirrors on the moon. See e.g the paper Progress in Lunar Laser Ranging Tests of Relativistic Gravity (Williams et. al. 2008) which established an effective limit on the expansion at AU scales that is about 80 times smaller than what would be expected if cosmological expansion applied within our solar system.

As John Rennie explained in an answer to this question, the expansion is a property of the FLRW metric, but the local distribution of matter doesn't match the assumptions for that metric (which hold well enough on cosmological scales). That doesn't prove by itself that a metric that describes our solar system doesn't have expansion, but the experimental evidence is that if it does it is much smaller than you'd expect from a simple extrapolation of Hubble's law down to AU scales.

Such measurements have been done, using lasers reflecting off mirrors on the moon. See e.g the paper Progress in Lunar Laser Ranging Tests of Relativistic Gravity (Williams et. al. 2008) which established an effective limit on the expansion at AU scales that is about 80 times smaller than what would be expected if cosmological expansion applied within our solar system.

As John Rennie explained in an answer to this question, the expansion is a property of the FLRW metric, but the local distribution of matter doesn't match the assumptions for that metric (which hold well enough on cosmological scales). That doesn't prove by itself that a metric that describes our solar system doesn't have expansion, but the experimental evidence is that if it does it is much smaller than you'd expect from a simple extrapolation of Hubble's law down to AU scales.