Responding to the bulk of your post separately from the last item about the blog argument, to reduce clutter:

Two free test particles each moving toward destinations beyond opposite sides of a LIF, as measured in a global frame containing the LIF, recede from each other as measured in that LIF.

This law is fine, but it doesn't apply in the case under discussion, because the two probes are not moving "toward destinations beyond opposite sides" of the LIF in question. I've repeatedly explained why not.

You disagree that any prediction for a global frame can affect experiments in the LIF.

No; the problem is that you are incorrectly translating predictions in a global frame into predictions for experiments in the LIF. I've repeatedly explained why your translations are incorrect. One example of an incorrect translation is that you think the two probes are moving "toward destinations beyond opposite sides" of the LIF. The correct translation is that the singularity is in the positive t direction in the LIF, while infinity is in the positive x direction; these are not opposite sides of the LIF.

Forget about black holes and light rays and especially complex GR terminology.

In other words, forget about the fact that the relationship between global parameters and local parameters is different for different LIFs, which is the whole point under discussion. The relationship between global and local is different for an LIF falling through a black hole horizon than it is for a "skydiver" frame free-falling at 1 km above the Earth's surface.

Let the analogous prediction for the global frame be this: any particle below 1 km above sea level must fall inexorably toward r = 0. Let your first probe be launched just above the 1 km mark. It doesn't need to be escaping, just always moving away from the Earth during our experiment. Let the second probe be launched just below the 1 km mark. As measured in any LIF containing both particles, they'll recede from each other.

If you set up the initial conditions that way, sure. But that already makes this experiment different from the one posed in the blog post, because the initial conditions in that one were that, as measured in the LIF, the probes were approaching each other.

As measured in a global frame, the first probe is moving toward a destination beyond the side of the LIF that's facing away from the Earth, while the second probe is moving toward a destination beyond the opposite side of the LIF that's facing toward the Earth.

Sure, because this LIF is at a radial coordinate that is way, way, way larger (about a million times larger) than the Schwarzschild radius corresponding to the mass of the Earth. So the relationship between directions within the LIF and directions defined globally is very, very different than it would be in an LIF that was free-falling just at the Schwarzschild radius. As long as you continue to ignore this huge difference, even though I've explained it multiple times and given you links to a pair of spacetime diagrams that illustrate it, you will continue to make the same mistakes you've been making.

For the record I still believe my argument using megaparsec-sized frames was sound. Your counterargument didn't explicitly show that any sentence in mine was false or didn't follow from its premises.

That's because I'm not going to hold your hand when making arguments; I expect you to use your intelligence. Your argument, as I remember it, assumed that you can independently adjust the size of the LIF and the initial velocities of the probes. You can't do that, because both of those things depend on the mass of the black hole. The dependence of LIF size on the mass of the hole should be obvious. The probe initial velocities depend on the mass of the hole because that affects what the escape velocity is at the point that the first probe is launched. So there is only one adjustable parameter: the mass of the hole. And my calculation showed how the LIF size is much, much smaller than the distance it would take for the second probe to catch the first probe, regardless of the mass of the hole.

(Edit: Having looked back, I see you also claimed, as far as I can understand your argument, that if the "catch-up" happens within the skydiver LIF, even if it doesn't happen within the astronaut LIF, that somehow invalidates the equivalence principle. That's wrong, and I explained why: the smaller of the two LIF sizes determines the range of comparison. If the "catch-up" happens within the skydiver LIF but not within the astronaut LIF, it must be because the skydiver LIF is the larger of the two.)