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the long period of the torsion balance will simply integrate the gravitational pulses into a continuous tidal force
G = 6.67384 × 10-11 m3 kg-1 s-2c = 2.99792458 x 108 m s-1∴ Gc = 2.01 X 10-2m4 kg-1 s-3This is far from the dimensionless number you quote, and if you use the fps system you get an entirely different figure before the decimal point.
You can calculate the reciprocal if you like, but it still won't be dimensionless, so the number has no associated magic.
Radiation dose is measured in joules per kilogram, which has the same dimensions (L2T-2)as c2 but it has absolutely nothing to do with the velocity of anything, nor with the latent heat of fusion which has the same dimensions.So I wouldn't consider dimensional analysis to be a necessary starting point for determining the speed of gravity, which will be in meters per second, whatever the number turns out to be.
The idea that the speed of propagation of gravity could be anything other than the speed of light is intriguing, especially if it turned out to be dependent on the local gravitational field, which is why a laboratory measurement, however inaccurate, would be fascinating at least, and mindboggling if it turned out to be variable.