The first probe moves toward infinity. The second probe toward the singularity.

Globally, yes. Within the LIF, it's impossible to tell, just from the way either probe is moving within the LIF, what is going to happen to it after it leaves the LIF.

Also, to the extent that "toward infinity" and "toward the singularity" can be defined within the LIF, they aren't the directions you think they are--more precisely, the latter isn't. See below.

They each move toward destinations beyond opposite sides of the LIF.

This is wrong and I've already explained why it's wrong, but I'll recap again since apparently you aren't reading my posts very carefully. The singularity is not in the negative x direction in the LIF; it's in the positive t direction, i.e., in the future. Infinity is more or less in the positive x direction, yes. But that means that both probes, as far as you can tell within the LIF, are moving in both directions: toward the singularity (in the positive t direction) and toward infinity (in the positive x direction). There's no way to tell, from within the LIF, where the probes will end up.

(You could, of course, define "toward the singularity" as "decreasing r coordinate" and "toward infinity" as "increasing r coordinate"; but as I have already shown multiple times, the respective probes satisfy those definitions while both moving in the positive x direction within the LIF; and as I've also said multiple times, the r coordinate is irrelevant to the LIF since it's global, not local.)

you wouldn't be able to show that the probes approach each other in the skydiver's frame

Wrong; I already have. But if it will help, I am working on a second spacetime diagram that shows the skydiver's LIF, to complement the one showing the black hole LIF that I already posted on PhysicsForums (see the link I posted in this thread). What that diagram will show is that the only difference between the two LIFs is how the lines of constant r coordinate appear; everything else is identical.

given the first probe moving toward higher r-coordinates and the second probe moving toward lower r-coordinates

This isn't true in the skydiver's LIF; it's only true in the black hole LIF. As I've already pointed out. And which is irrelevant to the equivalence principle anyway, since the r coordinate is global, not local. How many times are you going to repeat the same erroneous statements?

GR places additional conditions on what happens in a LIF, when it predicts that everything below a horizon falls toward the singularity

No, it doesn't. You can give a definition of what "toward the singularity" means in a given LIF (I did it above for the black hole LIF--note that it isn't what you think it is); but you can't restrict what happens within a given LIF based on which direction "toward the singularity" is. All a global definition like "toward the singularity" does is tell you the relationship between local conditions in the LIF and some global condition. That can help you to determine what the right local conditions are in an LIF to model some desired global condition; but it can't ever tell you that a given local condition that would be permissible in special relativity is not permissible in an LIF.

we must be able to set up a thought experiment that accounts for that prediction within an LIF straddling the horizon.

Yes, and I've already shown, multiple times, how the thought experiment we've been discussing does this.

For example, it's okay to have an experiment in an LIF that says "fire a particle toward Vega", even though Vega is outside the LIF, as long as the measurements take place wholly within the LIF.

Sure; and the way you would model that experiment is to determine what condition in the LIF corresponds to the global condition "fire a particle toward Vega".

Similarly, in the thought experiment under discussion, the way we model it in the LIF crossing the horizon is to determine what local condition in that LIF corresponds to "the first probe is fired at escape velocity just outside the horizon", and "the second probe is fired at a higher velocity, relative to the astronaut, than the first probe, but just inside the horizon". And then we have to check that the two probes' trajectories, in the LIF, that we come up with meet the criteria of "first probe is moving toward infinity" and "second probe is moving toward the singularity". And I've already shown, multiple times, how all these conditions are met, and how your claims that they are not met are based on an erroneous understanding of how the global conditions translate into local conditions in the LIF.

At this point, I'm tempted to conclude that you're not really trying to understand my posts, but just looking for things to disagree with based on the erroneous straw-man version of relativity that you have in your head. You keep on repeating the same errors, even though I've already shown why they're errors, and you keep on missing the things I'm telling you about how relativity actually models this scenario, that address the concerns you're raising. I really think you need to step back and take an honest look at what I've been saying.