An important question; by what process does this missing Doppler shift disappear?
All rest mass and rest energy increases with velocity whither you use Newton's of Einstein's equation for kinetic energy. That includes mass in a molecule that will be released as a light photon. There there is a missing Doppler shift in all directions its frequency and energy ratio =m/mo=1/[(v/c)2-1]0.5. Note this is reciprocal of the transverse Doppler shift and frequency x h=energy for photons.
By what process does it disappear?