To motivate my question, I'll share an example from Carroll. Carroll shows that if we have a simple model of a star as a perfect fluid characterized by some coordinate-dependent pressure $p$, energy density $\rho$, and radial coordinate $R$, the Schwarzschild parameter $M$ outside of the star is simply $\int_0^R \rho(r) 4 \pi r^2 dr$. However, Carroll notes that the integrated energy density $\bar{M}$ using the induced metric will be $\bar{M} = \int_0^R \frac{\rho(r)}{\sqrt{1-\frac{2G}{r}\int_0^{r}\rho(r') 4\pi r'^2 dr' }} 4 \pi r^2 dr > M$. He intuits the difference as the effect of negative gravitational binding energy. Note that the weak-field limit of the Schwarzschild metric shows that a faraway observer would feel the effects of the star like a Newtonian observer near a star of mass $M$.
My point of interest is that we will suffer different $M$ values for a fixed $\bar{M}$ (and vice-versa) if we change how the energy is distributed within our simple model of a star. That is, I interpret this result as that we no longer enjoy "Gauss's law for gravity." (Ultimately, I believe the energy density will be decided by the requirements of hydrostatic equilibrium and the introduction of some equation of state relating pressure and energy density, as discussed in Carroll).
On stackexchange, users like Rob Jefferies have noted that we might expect "a significant reduction in the gravitational mass compared with the baryonic mass" for a neutron star.
Now for my question. Consider a black hole, where far from the black hole in the weak field limit an observer feels the effects of gravity like that from a Newtonian object of mass $M$. I'll call $M$ the faraway mass. (Though Schwarzschild has a clear definition of mass $M$ through the metric and through e.g. the Komar mass, I would prefer to consider the mass $M$ in terms of the leading order geodesics in the weak-field limit far away from the black hole. )
Are the above considerations of the distribution of energy density important in finding the value of $M$? If I have some relatively localized, relatively small energy density corresponding to some total energy $m$ far from the black hole and it drifts into the black hole and enough time passes to reach some equilibrium metric, should I expect that my observer will (eventually, after some time) feel a faraway mass of $M+m$ or a faraway mass less than $M+m$? (I expect less because of radiated energy.) How can I determine the faraway mass my distant observer will feel - do I need to solve all the intermediate dynamics in the style of an initial value GR problem and find the radiated energy, or can I use some other method to extract out just the final faraway mass?
