What single quantitative test would confirm a stationary Earth while explaining parallax, Doppler shifts, CMB alignment, and tracking?

You already have several tests that confirm it. The Michelson–Morley and Michelson–Gale experiments, the Sagnac effect, and the cosmic microwave background alignment all point to a stationary Earth. The parallax and Doppler shifts fit perfectly if the heavens move around us. The CMB has an axis that runs right through Earth. Heliocentrism has to twist relativity to explain that away.