Good idea, you could do a monte carlo method where you throw many packing peanuts at the cat (or many cats!) and count how many stick, divide by how many you threw, et voila

Why would the universe care about dimensional analysis? Besides, the outside constant would do the unit conversion from whatever the right-hand side produces to units of force.

you have free constant k in front of the equation

any dimensional analysis gets consumed by its unknown dimensionality

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q^2n = (-q)^2n which know is ruled out by experiment.

doesn't mean equation can't be using absolute values ("number of electrons/protons") and just applying needed sign at the end

Supposedly, one way to make two equal charges is to charge one thing and then put it in contact with the other thing, so that the charge splits by symmetry. But then how would we check that indeed we have two equally charged things?

same as with making guaranteed flat surface - you make 3 and measure each pair

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

Could you use an electromagnet and a metal ball to create a fixed charge? If you change the current in the electromagnet, that would change the intensity of the magnetic field. With the metal ball fixed in place nearby and a careful curve for the change of current, you could potentially have a constant charge remaining on the ball.