This is an excellent question.

Light exiting from stars deep in a galaxy (which is in turn deep in a galaxy cluster) is subject to gravitational redshift. The greater the mass of the galaxy and cluster, the greater the redshift.

If we add transparent, non-radiating mass to a galaxy and its cluster, the mean redshift of starlight from the galaxy will redshift, and we'll see the result on various absorption and emission lines in the spectrum.

Note, we do not need dark matter for this; increasing the central mass of a galaxy cluster will do the trick too. We can do this by adding more stars, or by adding black holes, or by making a central black hole more massive. All of this will cause an increase in gravitational redshift.

However, adding more central mass to galaxies will be reflected in the trajectories of clouds of gas and collections of stars within the galaxies; likewise, adding more concentrated mass to a galaxy cluster will alter the trajectory of galaxies within it. In general, these objects will have faster orbits -- fast enough that a non-gravitational redshift will be apparent in the arrangement of spectral lines. (We would also need to figure out how to keep any central masses from being extremely bright anywhere in the electromagnetic spectrum, and how to keep them from eclipsing parts of their home galaxies and clusters that are behind them from our perspective.)

If we instead redistribute our increase in galactic mass into a gaseous halo-like structure that extends well beyond the outer stars, we get the increased gravitational redshift without requiring stars and gas to travel faster to avoid falling inwards. The halo still has to be electromagnetically quiet, and non-eclipsing, or we would see it. (This all applies to clusters, too).

Stepping away from the "dark matter puzzle" of the spectral lines of molecular and atomic gas clouds in galaxies and clusters of them, we can then look at another aspect of your question.

Galaxy clusters, from their point of view, are just floating passively in space.

The expansion of the universe follows a model wherein if we treat the astrophysical content of galaxies and clusters as sensitive accelerometers that in aggregate show off the peculiar accelerations experienced by these large scale objects, there is no evidence that these accelerometers should report a non-zero magnitude or point in any particular direction. That is, galaxies and galaxy clusters are themselves in free fall, and we can see this because of lines of evidence including expansion-direction-and-magnitude squashings of https://en.wikipedia.org/wiki/Light_echo lines turn out to be null, likewise there are no radio-loud edge shocks for galaxies that appear to be accelerating away from us, there are no shape distortions of the most highly "accelerated" distant galaxies, and so forth. There is simply a spectral redshift.

So, for an isolated galaxy the spectrum it emits depends on a combination of its starshine, any noise from close to its central black hole(s), the noise from any thick gas clouds undergoing gravitational collapse, and the absorption and reemission of diffuse gas clouds heated by these types of emission and absorption-and-reemission sources and the distribution of dark matter within and around the galaxy. The more dark matter, the more redshifted the galaxy's spectral lines will be.

Preserving the Copernican Principle, immersing a set of such isolated galaxies into an expanding cosmos leads to the inhabitants of each galaxy observing a cosmological redshift on the gravitatonally-redshifted spectrum emitted by other sufficiently distant galaxies. The gravitational-redshift depends on the mass of the galaxies including their dark matter. The cosmological redshift only depends on a light-travel distance from the galaxy, no matter how light or how massive that galaxy (or its parent cluster) is.

Next, if we have an arrangement b-->F-->U, where b is a background galaxy, F is a foreground galaxy, and U is an observer, we will that the light from b is cosmologically redshifted before it interacts with the molecular and atomic gas clouds in F, causing a spreading of absorption and reemission lines rather than a thickening. For the very obvious lines from neutral atomic hydrogen, we see this as the https://en.wikipedia.org/wiki/Lyman-alpha_forest which is an important tool in astrophysics and in checking the soundness of our theory of gravitation.

We would get a Lya-forest structure even in the absence of dark matter in b or F or anywhere else; only the fine strucutre would differ. Indeed, we could substitute a diffuse static cloud of atomic hydrogen for F and still see a "forest" structure minus small contributions from the starshine in F and the orbital motions of the gas in F around some central mass. The deep infrared noise from F in that case, however, would not be gravitationally redshifted.

Finally the answer to your question: the cosmological redshift does not in general depend on the "late" distribution of dark matter. (Late as in at least millions of years after the hot big bang). But the gravitational redshift on light emitted by stars and galaxies does depend on dark matter and in general the starshine is reddened by it. The reddening is because of the additional mass of the dark-matter-containing structures, and also because the distribution of the dark matter into halos allows for slower orbits of stars and gas clouds, thus suppressing their non-gravitational non-cosmological redshifts. (In an edge-on spiral galaxy without dark matter we would see a greater difference in the blueshift of one extreme tip of the thin disk and the redshift of the opposite extreme tip of the thin disk, compared to the same spiral galaxy with dark matter; the light-and-radio from the central bulge would be less redshifted in the absence of dark matter).

So, no, the cosmological redshift does not depend on dark matter. The gravitational redshift on light from galaxies, however, does. So does the non-gravitational, non-cosmological redshift on gas clouds whirling around the edges of rotating (e.g. spiral) galaxies, or rising and sinking on radial orbits within elliptical galaxies.