I'm not sure we disagree. Cosmological redshift doesn't rule out additional changes in observed frequency that compound due to the ones we observe by the expansion of the universe. Hubble originally explained his observed redshifts in other galaxies entirely due to the doppler effect but changed his mind when he observed a correlation between the distance of a receding galaxy and its redshift. If the doppler effect could explain cosmological redshift then galaxies would seem to have to 'know' how far they were from the observer!

If you go back and see the original 46 galaxies that hubble plotted by comparing observed redshift to the distance hubble had observed you can see considerable 'scatter' in the observations. This scatter is due to the Doppler effect.

The velocity distance relationship in hubble's law isn't due to the doppler effect but a model that relates recessional velocity to the general expansion of the universe.