I went to give a similar answer and realized that - in the instance that the two galaxies are sideways facing each other and with the text saying “it gets brighter” not “one side gets brighter” - I’m not sure if this is the actual answer.

I can’t figure out why the Doppler Shift would make the entire galaxy brighter. I assumed it would make the rotation side spinning towards us brighter. But also redshift the side spinning away.

How are you linking that effect to the spin orientation of the Milky Way galaxy? Where is the all-sky anisotropy coming from?