The nature of the photon, the carrier of the light
The nature of the photon, the carrier of the light, is such that a fast moving light source doesn't actually alter the 'color' of the light. The fact is easily recognizable.
The color of light is determined by the energy environment within the electromagnetic photon envelope, when the light is created. Some photon packets are larger, some are smaller. The higher the energy level is, inside a photon package, the tighter the package is held together, and the smaller it thereby becomes. The physical movement of an atomic element in which a photon package is formed, does not affect the size of the package. Only the energy-level within the atom affects the size of the photon and thereby its color. Different size photons are recognized as different colors. This fundamental pattern, of course, extends far beyond the visible spectrum, which is actually quite narrow. Extremely high energy levels, for example, produce extremely tiny packets, such as the x-ray 'photons' that are typically 100,000 times smaller than the photons of visible light. But why won't light change its color, once it is created? Let's take a look at that.