It's probably easier to verify Gauss's Law, which then gives you Coulomb's law for free.

Direct verification of inverse square laws is hard! For electromagnetic interactions (ie Coulomb's law) you can use scattering. If you want to do a static experiment (Coulomb's Law the hard way or gravity) you probably need a torsion pendulum experiment. AFAIK the best in the world at that are at UW in the Eöt-Wash group: https://www.npl.washington.edu/eotwash/torsion-balances