"When we use the Schwarzschild geometry, we have to remember from what perspective the equations are written."
The terms of the equations make this perfectly clear.
"The COBE experiment indicated that the background radiation was essentially uniform with small
variations in angular distribution. This may be interpreted as our neighborhood of the universe being in equilibrium with this background radiation. This would be the local frame of reference."
The CMBR is a global distribution of remnant photons with a black body spectrum. These remnant photons are everywhere, have black body spectrum and are invariant in coordinate transformations.
The local frame of reference is the shell frame and it is 'just that' local. The term shell comes from the Schwarzchild geometry which has a spherically symmetric non-rotating object (such as a Schwarzchild black hole) at the origin of the coordinate system. Theoretically a shell could be constructed at any radius from the origin which would have all points on the shell measuring the same distance to the origin. All shells would be spherically symmetric and only differ in r. The local shell observer would be stationed at the shell (A) and evaluate events such as the speed of a stone falling from infinity as the stone crosses shell (A). The local shell observer always uses the metric of special relativity (the metric of flat spacetime) to evaluate local events. The Schwarzchild metric describes the far away observer (Bookkeeper) perspective. Both shell and bookkeeper perspectives are equally valid. They both measure what is observed. The Schwarzchild geometry can't differentiate between different types of energy such as potential and kinetic. When you do the physics from the bookkeeper perspective you look at conservation of energy in a global way. You look at the entire coordinate system which for this case would be the stone falling from infinty and the black hole.