The mass difference between a bare proton and a bare neutron is 1293 keV, while the mass of an electron is 511 keV. That means that the effect you're talking about is often going to be on the same order of magnitude as the difference in mass between an atom and one of its ions. In something like liquid-phase chemistry, as opposed to mass spectroscopy, I doubt that an ion ever really even maintains its ionization state for long enough to make the electron-mass correction even meaningful, and if that's not meaningful then the correction you're talking about, which is on the same order of magnitude, is also not very meaningful.
When you want to start talking about corrections at this level, you probably need to define more carefully what purpose you're going to use the resulting numbers for. For example, if you're going to look at the infrared spectrum of the NaCl molecule in a gas, which comes from its end-over-end rotation, then it's got some moment of inertia that is mainly determined by the nuclei, but when the bond forms, one of the electrons shifts its mass around, and that will change the result.
Keep in mind also that, conceptually, the protons and neutrons in a nucleus aren't really the same creatures as the particles that exist in free space -- the "bare" proton and neutron I referred to above. A free neutron is a little ball of quarks and gluons, with pairs of particles appearing and disappearing spontaneously in a kind of quantum soup. The mass that we measure for the neutron is some kind of average or composite value that represents the masses and kinetic energies of all those sub-particles. When a proton or neutron is incorporated in a nucleus, it loses its individual identity, and it's no longer really a proton or neutron. It promiscuously shares all its quarks and gluons with its neighbors. It's only a kind of shorthand terminology when we refer to the nucleus as being made of protons and neutrons. So it's not even really totally obvious whether it makes sense to try to add the masses of all "the protons and neutrons" inside a nucleus. It does sort of work as an approximation (it's what's done in standard versions of the nuclear liquid drop model), but it's only an approximation.
Another thing to realize is that most chemistry is done with light elements, for which the proton number and neutron number are about the same. Since the atomic mass standard is based on 12C, which has equal neutron and proton numbers, you will only see any effect in a.m.u. masses when you go to heavy nuclei.