the assumptions are otherwise in line with the 1E PHB years ago. So yeah, bog-standard paladin type, which has subsequently been fleshed out as time has gone on.
The problem is that the 1e Paladin as specified in AD&D had almost all of its assumptions unstated, except for a few mechanical things like 'doesn't use poison', 'must employ good aligned henchmen', and so forth.
Bright Justinian, The Hammer, The Clear Eyed, Who Brings Light into Darkness, God of Justice, Fairness, and Truth is my 1e 'god of Paladins', before my views evolved considerably. As such, most of what I say with regard to followers of Justinian applies here.
1) If you enter into a treaty with this brigand in the first place, then in doing so you are recognizing her legitimacy as a someone who can be lawfully treated with. As such, you are de facto not holding her past crimes against her as something which requires justice. You are essentially saying, "We can't justly address what has gone before this point, but so that this situation will not persist, let's clarify what just dealing will be between us in the future." or perhaps, "We mercifully forebear from holding you accountable for your past crimes, provided you enter into this agreement." You enter into that sort of situation when the past is complicated, restitution is hard, and attempts at justice now will only create further injustice and suffering. Fine. It might not be wise, but it lawful and good in intention.
But if you do that, you can't decide to hold those past crimes against her now unless she's broken the agreement. What evidence do they have that she's broken her agreement? What evidence is there that she hasn't been trying to uphold her end of the bargain? What evidence do they have that she is responsible not for crimes generally, but this specific crime? You can't go to court without charges, and you can't prove a specific charge by saying, "Well, we know this is a bad person." You have to prove the guilt of this specific crime, that she broke her agreement. Because if she didn't, the guilt is with those that did, and not her. The situation would be like holding a president accountable for warcrimes committed after the president was disposed by a coup, imprisoned, and a new regime was formed. Even if the president wasn't a nice person, even if the treaty was unwise, you can't hold the wrong person accountable.
In this context, executing the half-hag seems a lot like vengeance and not justice.
2) You can't execute someone for their future actions. That's not justice. You could hold her accountable for conspiracy to commit various evil deeds, if you could prove that she was not only planning such deeds, but had already taken concrete steps toward implementing them. But that's again prosecuting someone for something that they have done, not something that they might do in the future. So, with neither past or future deeds justly on the table, it's not at all clear what they could fairly try this person for. Being tied up? Being in the wrong family? Being ugly? Not being nice enough to the PCs?
3) The 'crime' here appears to be to have been tied up and been at the wrong place at the wrong time. In a strange sort of way, her ogre brothers have better followed the dictates of justice than the paladin has. How many times in movies does the vigilante leave the henchmen alive, but tied up and helpless, awaiting the judgment and mercy of someone else - presumably someone more ruthless than themselves. Now, I'm not saying a Paladin can be held to such a low bar as allowing evil to dispense justice, but the ogres appear to have done this very thing - they left the villainess who entered into what they considered a bad agreement to be dealt with by the Paladin, presumably because they knew that the Paladin was even more ruthless than they were, so that they could say that they wouldn't have their sister's blood on their own hands. When ogre butchers are teaching you lesson is justice, you probably aren't acting like a paladin. Your standards have fallen abysmally low.
4) You can't lawfully kill someone without trial except in self-defense. The prisoner was helpless. This wasn't self-defense. It wouldn't have represented an undo burden to cart her back to town and hold a fair and public trial, and even if it did, you can't just execute someone because giving them a trial would be burdensome. Maybe the legal authorities wouldn't have recognized the paladin's former agreement (in which case, if it wasn't a legal agreement, why did the paladin agree to enter into it), but that at least wouldn't have been the paladin's fault. He would have been accepting the judgment of his superiors. Is this paladin acting like someone that is used to accepting someone else's judgment, or is he acting (chaotically) like he is the source of judgment?
5) For crying out loud, if you could exercise mercy back when she was leading a band of brigands, what prevents you from exercising mercy when she is tied up by one? These actions make it look like mercy was never the issue, but rather that earlier they entered into the treaty because they feared her, and now when they don't fear her, they kill her. That isn't justice. That isn't idealism. That's a pragmatic cold-blooded calculation more suited to a Neutral character than one that is Good. It's almost like being good and doing what is right a secondary consideration compared to doing what is easiest and going with the flow. That doesn't sound like a paladin to me.
If you are serious about doing this the 1e way, Paladin to me 50/50 has just lost Paladin-hood depending on information I don't have about the villainess - did she have a legitimate grievance (was the sword stolen from her family?), was she capable of reform if treated kindly, is she still lying, is this behavior by the paladin part of a general disregard for the law, and so forth.