Since you're keeping it short, I will as well. I don't think it's worth discussing the blog post because it's too vague and it makes too many misstatements about what relativity says. That's why I tried to take out what I saw as the essential points and put them into puzzles that were properly stated. If you can't or won't discuss the puzzles as I presented them, I won't be able to give much of a response. All I can do is point out some particular items that seem to me to capture the mistakes you and the blog post author are making.

The cloud of particles is demanded by GR to be splitting in two along the horizon

This is wrong; GR "demands" no such thing. The blog post's author has misunderstood GR in this respect. See below.

When you talk about "global coordinate chart" and a "lightlike curve" you're being unnecessarily complex, I say

And I say that you are making a big mistake here, because a proper understanding of those concepts and how they relate to the LIF centered on the horizon is crucial to properly understanding and stating what GR says about this scenario. You'll see several examples of this below.

one basic prediction of GR as it relates to the horizon, namely that everything below the horizon moves inexorably inward toward the singularity

That means everything below the horizon must decrease its r coordinate in the global coordinate chart. It does not mean that everything below the horizon must decrease its x coordinate in the LIF. Remember: the horizon itself is the line t = x in the LIF; i.e., the horizon is moving in the positive x direction in the LIF at the speed of light, and it passes through the origin of the LIF at t = 0, x = 0. So a "cloud" particle that is at some negative x value at t = 0 in the LIF will be inside the horizon, and a "cloud" particle that is at some positive x value at t = 0 in the LIF will be outside the horizon; and it is perfectly possible for the first particle (inside the horizon) to be moving in the positive x direction in the LIF faster than the second particle. And this can be true even if the second particle (the one outside the horizon) is moving at "escape velocity". And it can also be true even though the first particle's r coordinate is decreasing and the second particle's r coordinate is increasing. My previous post gave more details on why those statements are true.

Oh, and one other thing: what does "toward the singularity" mean in the LIF? Which direction is the singularity in? The answer is, it's in the positive t direction--i.e., it's in the future. So any object in the LIF is moving "toward the singularity" as far as the LIF is concerned, since all objects move towards the future. There is no way to tell, from within the LIF, which objects are going to ultimately hit the singularity and which ones are not. That's a global concept.

To prove the blog wrong you need to show that the cloud doesn't split in two, that all the cloud's particles can in fact move outward in formation.

My previous posts (plus all the ones in the earlier HN thread I linked to, plus what I posted on PF) have already done that, multiple times. However, I've given a quick recap above.

This discussion can be much simpler than you're making it.

If only that were true.