Talk to Sagacious Zu more
That's the short answer; note that the longer answer below is riddled with spoilers.
Basically, in the playthrough that I had just completed when I wrote the question, I hadn't been paying much attention to this companion. Whilst looking for something else, I stumbled upon the answer on the Jade Empire wiki, but rather than just link that, I did another playthrough to prove that this was the answer to my question, so below is my findings in more detail (note that only number 1 is what you have to do to "unlock" this other content, the other numbered points are just my description of what happened next).
1. Make sure you talk with Sagacious Zu before he leaves the party
Throughout Chapters 2 and 3, you can talk to Zu (either as your active follower or at camp, it makes no difference) to learn more about his past, but after each conversation he won't say more until you complete another part of the main quest so that his past is drip fed to you over the course of the game; this is how it works with all of the companions' backstories. Note that, in Zu's case (and the romance-able followers, not that that's relevant here), it is possible to say something dismissive that effectively causes them to never speak with you again about their backstories, so basically don't pick the last option if it sounds dismissive or rude.
However, he will leave the party permanently during Chapter 3, specifically when you hand in either of the two Lotus Assassin recruitment quests, so you must have learned all you can before doing that. The final conversation is the one during which one of your responses is "Enlighten me.", so if you get to a point where you can say that to him, you have reached the last conversation you need and can safely hand in Lotus Assassin recruitment quests after that point.
(In fact, I missed this one during the playthrough I was doing for this answer, and had to reload an autosave in the Arena, then make him my active follower and talk to him in the Arena before I left the Arena, because depending on how you complete these recruitment quests, it makes you return to your followers and advance the plot as soon as you leave that building, so having Zu be your active follower and talking to him before you leave the Arena might be the only way to have this "Enlighten me" conversation).
2. Zu tells you Master Li's child is alive in the Lotus Assassin's HQ
During late Chapter 3, you infiltrate the Assassin's HQ and work for them for a bit, and Zu keep approaching you and slipping off again, offering you advice. When you eventually kill the Assassin leaders Shin (and possibly Gang), then upon leaving the Golem Press Room, then if you did number 1 above right, he should say more than he usually says at this point.
Usually he tells you that he was one of the Lotus Assassins who was ordered to murder Master Li's family, but this time he will add that he hesitated, then turned on the other Assassins after they killed Master Li's wife but before they killed the baby, following this up with "Master Li's child still lives". This is still only a hint, but it is the first point where you can safely say that the "learning about Master Li's child" subplot has been advanced (if Zu doesn't say this, you'll have to load a save from earlier, before Zu left the party, like I had to).
3. You tell your followers what Zu told you
At the end of Chapter 3, after escaping from the Assassin's HQ, you return to your followers and can at that point tell them that not only was Zu one of the Assassins sent to kill Master Li's family, but that the child was not killed. Specifically, you can say "There is more. Zu told me about Master Li's murdered family. The child is alive.", which is usually slightly differently worded and omits the last part if you didn't advance Zu's plot.
I'm not sure if it matters whether or not you choose this option, telling you party, or choose the other option and gloss over it, not telling your party; I don't know whether it still comes up later or if picking the other option basically cuts the plot off at that point, so if you care about this, don't pick the other option, I guess.
4. The Water Dragon tells you that it's Dawn Star
The next part of this plot you won't learn until the end of Chapter 5, when the Water Dragon tells you that she can sense that you have knowledge of Li's child, and then gives you lots of hints from which you can only conclude that it's Dawn Star (she pretty much spells it out without actually saying "Dawn Star" herself, although one of your responses includes "Dawn Star", so it's a given at this point).
5. Telling the party that it's Dawn Star
During Chapter 6, when you reunite with your followers, an extra chunk of dialogue exists that isn't there if you didn't learn of Zu's backstory. Dawn Star will confess that the vision she had that led them to Dirge to meet you also revealed to her that she is Master Li's daughter, although she kept it to herself because she couldn't quite believe it, but says it now in front of you, and you can confirm.
Silk Fox isn't very convinced, but apparently isn't too upset about this because, if you are trying to romance (it's a BioWare game, of course there's romance) both Dawn Star and Silk Fox and choose the option to romance both of them, they're happy to have a threesome with you, even though they just found out that they're cousins, which is a bit creepy, to say the least... at least it mercifully just fades to black at the end of the conversation and shows nothing.
6. Telling Master Li that his daughter is Dawn Star, his old pupil
At the end of Chapter 6, when you confront a projection of Master Li, you can at that point tell him that his child lives and that his daughter is Dawn Star. At first, he's angry and insulted, but then he considers it seriously, only to dismiss it because he never really cared anyway. Ultimately fruitless, but still an interesting extra piece of dialogue.
7. Telling Dawn Star about Master Li's reaction
As Chapter 6 passes into Chapter 7, you can talk to your followers (I'll point out that this was an Open Palm playthrough, so no enslaving Death's Hand and half the party; Dawn Star's reaction might be different to what I've written here otherwise, but I wouldn't know). Dawn Star specifically requests to talk to you (which she doesn't otherwise if you didn't learn of Zu's backstory) and you can tell her that Master Li didn't seem to care, which she is disappointed by. I'm not quite sure how she knows you spoke to a projection of Master Li, but that's what happens.
8. Dawn Star can confront Master Li herself
Near the end of Chapter 7, as you finally confront Master Li in the flesh, if you have Dawn Star as your active follower (the top of the stairs after fighting all the Assassins before you see the Water Dragon in the flesh is the last chance you have to swap followers, by the way), then she will ask you if she should talk to him, and if you let her, she tries to appeal to him, but he ultimately dismisses her.
If Dawn Star is not with you (or perhaps if you told her not to talk to him? I didn't try that specific outcome), you can instead try to appeal to him with regards to Dawn Star, but he is equally dismissive; the only real difference is who he addresses, but the dialogue is slightly different. In either case, he ultimately doesn't care, and it still ends with the same boss fight, but this is how the last part of this little subplot of the game plays out.
