The cosmological redshift can be also thought of as a Doppler shift. We cannot distinguish the difference between cosmological redshift and Doppler redshift, by just looking at the redshift data.
For example, let us imagine that you are looking at a car on the road. And you see that car is going with 50 km/h. You can say, the car is moving with 50km/h naturally. Or, I can claim that "the road is expanding" in such a way that car seems to move away with 50km/h. And by just looking at the light coming from the car, you cannot say which one is true.
Simply, the redshift of the galaxies can be thought of as a Doppler shift or cosmological redshift, it just depends on which frame you want to choose.
The CMBR data actually can contain the doppler effect since the COBE satellite (A satellite that measured the CMBR) rotates around the earth, the earth rotates around the sun, son rotates around the center of the milky way etc.
So the original CMBR image also contains this Doppler shift affect but we can remove these effects from it.
CMB dipole distortion
Reference for the image
After removing these affects we get the normal CMBR image.