The Wikipedia article on Moseley's law seems to show that the screening of heavy atoms is by 1 electron charge exactly (in the limit of large Z, experimental precision, within nonrelativistic limits, and).
But why is this exactly one unit? The other K-shell electron is not screening exactly one unit, and this seems to be a conspiracy of other electrons. I suspect it is because of an unappreciated hole-picture of deep holes in heavy atoms (electrons missing in deep shells), and I will describe this theory briefly.
If you remove an electron from close to the nucleus, the electron-hole behaves as an object with positive charge and negative mass (this is why it orbits the nucleus that it is repelled by). The state is not a vacuum quite, because of the presence of other electrons, but the rigidity of the Fermi liquid near the nucleus of a heavy atom means that the hole behaves as a single particle. This single-particle behavior is in the potential background of the nucleus and the other electrons, and it is possible that the result can give an exact 1 unit screening. I developed the formalism a little bit to see what the form should be, but I did not see any reason for 1 unit screening. Perhaps there is none, but it looks to be more than a coincidence.