The Doppler Effect is usually thought of as a source moving and its effect on the wavelength received at a frame of reference that is stationary. The same happens when the reference is moving and the source is stationary. The apparent wavelength elongates under both scenarios.
Assuming that the galaxies are stationary, the stars in a distant galaxy revolving toward us would have their light shifted with shorter wavelengths. But any radial velocity of the stars caused by attraction toward the center of their galaxy (black hole or not) would diminish the effect. On the other hand, stars moving away from us would have the wavelength elongated further by the same radial velocity.
When we measure the wavelength of light from a galaxy, you measure the summation of all the light we are receiving from the galaxy and we are not stationary. You receive more light from the near side of the galaxy than you do from the far side since the density of photons is decreasing the further they travel from a point source and the sources on the near side are closer. The further we are from a galaxy the more pronounced this effect would be. Thus the light from the near side would produce more waves and a more apparent elongated state at our stationary receiver. If the receiver is our planet and is moving toward the center of our galaxy, the effect would be intensified even if the galaxies were stationary.
To answer your last question, I don’t have a reference to cite since I’ve been unable to find any reference that discusses the situation. So I simply question the use of Doppler and put forth a hypothesis that the results may be biased. I’d enjoy reading any reference that you know which discusses the situation as I described.