I'm confused: cosmological redshift and doppler redshift are different phenomena, no? One measures the relative speed during emission, the other the total expansion of the universe since the emission. In GR far away galaxies don't really move away in space, but the in between space expands according to the friedmann equations. Or do you claim that both are equivalent?
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition