Is there any way to distinguish the doppler redshift from a distant galaxy, caused by its initial recessional velocity relative to this galaxy, from the probably much greater, cosmological redshift, caused by the expansion of space?
The two effects are observationally indistinguishable for an individual galaxy. The redshift associated with universal expansion is often interpreted as a recession velocity and there is nothing in a spectrum that tells you what fraction of the redshift is contributed by which effect.
If a distant galaxy is associated with a cluster of galaxies, then you can make an attempt by assuming that the average recession velocity of the cluster represents the cosmological redshift and that the perturbations from that for individual galaxies are their peculiar velocities.
Nevertheless, the cluster will have it's own peculiar velocity as an additive term to all these individual peculiar velocities. Fortunately, that doesn't really matter when it comes to studying cluster dynamics.
Since the cosmological redshift increases with distance (Hubble's law), whereas peculiar velocities do not, then at large distances cosmological redshift can be assumed to dominate. See What does "located in the Hubble flow" mean?