Context: Harry's dark side is amoral, destructive, will take any available option which leads to its target no matter how it may escalate or what the risks are, and cares about nothing else other than achieving regular Harry's current subgoal. (I'm convinced Eliezer regards the dark side as basically a UFAI.) Emphasis added:

...Harry plunged himself into his dark side...offering his dark side anything if it would only solve this problem for him

Who are the major players here that Harry can affect? Harry has no hold on the Wizengamot, as I pointed out any threat on Azkaban is more easily dealt with by attacking Harry.

So Dumbledore and Lucius are the keys. What can Harry do with Dumbledore - no matter the cost to Dumbledore, Harry, or anyone else - that would free Hermione? There's little he can testify to, as an Occlumens, so he can't even sacrifice himself (Lucius would refuse it), and it's not obvious how any of his magic 2.0 abilities could somehow convince the Wizengamot that Hermione is innocent or Lucius to let her go - what is he going to do, promise some more magic to an aristocrat who can buy all the magic he wants?

The answer is so obvious I'm surprised that no one seems to have suggested it yet here or in the reviews: Harry can use his leverage on Dumbledore to trade him for Hermione - "tradeoff". Dumbledore practically says as much:

"I would not do that to you," the old wizard said, a terrible weariness seeming to suffuse him as he turned to go. "Still less to Hermione. But I have no rabbits in my hat, Harry. We can only see what Lucius Malfoy wants."

What does Lucius want? Well, he is perfectly clear:

When Lucius Malfoy spoke again his voice seemed to tremble ever so slightly, as though the stern control on it was failing. "Blood calls for repayment, the blood of my family. Not for any price will I sell the blood debt owed my son. You would not understand that, who never had love or child of your own. Still, there is more than one debt owed to House Malfoy, and I think that my son, if he stood among us, would rather be repaid for his mother's blood than for his own. Confess your own crime to the Wizengamot, as you confessed it to me, and I shall -"

Why would Dumbledore do it? Because he's already half-way to turning himself in (viz his little dialogue with Madam Bones):

...The old wizard stood at the podium, his face twisting, untwisting - ...

and he really thinks Harry is on the path to darkness (in a way few others are, because Dumbledore is one of the only knowers of the Prophecy) and this Hermione incident would be more than enough to turn Harry, convince him that the system is irredeemably corrupt and turn his mission to 'taking over Magical England', as indeed the omniscient narrator tells us Harry has already done to the point of no longer caring about not being called a Dark Lord... Sacrificing himself to keep Harry on the side of good is a good deal. This is consistent with canon Dumbledore losing power and respect, and ultimately dying in the war with Voldemort while working on the Horcruxes to aid Harry's ultimate victory; and for that matter, who replaces Dumbledore as headmaster in canon? A character which just showed up in MoR for the first time ever...

It will come at a major cost - Dumbledore will either be in Azkaban or he will flee or something like that and his entire faction discredited. "Tradeoff".

To me, this is the most compelling scenario, which I give a full 40% probability of having; but I also like the debt (30%/20%) and time-turner strategies (35%), although the latter is more because time-turners are so general and powerful that I have to assume my inability to think of a really solid strategy is my inability alone.

A man with a scarred face sitting next to Lucius; Fenrir Greyback?

Wasn't that one of Lucius's lackeys from the previous chapters where they watched the battle?