So I’ve seen many, many AU fanfics where the basic premise is that Zuko and Azula have a much better relationship than canon, and these usually end up involving conspicuous amounts of patricide. I have my issues with this premise, but it can be interesting or entertaining if done well.
What I’ve never seen done is an AU where the point of divergence is that Azula, Mai, and Ty Lee have a much better relationship than canon. You could do this by having the Dangerous Ladies meet and bond much younger than they did in canon(I’m talking as toddlers here, since I’m guessing they became friends somewhere between the ages of 4 and 6 in canon. Mai’s a year older than Azula). You could have so that at some critical moment in Azula’s life, she really opened up and unloaded her trauma upon them, and they supported her and in turn unloaded their traumas to her. You could have any number of points of divergences. But the ultimate point would be that, at some level, Azula loves her friends more than she loves anything else in the world. More than her father who is more always so demanding of her. More than her mother who abandoned her and always favored Zuko anyways. More than her entitled, jealous brother who can’t stand that Azula’s better than him. Quite possibly more than she loves her nation, her duty, and her status as princess. And, at some level, Mai and Ty Lee reciprocate and love each other and Azula very, very deeply.
In fact, for a canon divergence the simplest thing to do would be to have Mai and Ty Lee never leave Caldera(after Zuko’s banishment). Let’s say Mai’s father never receives a post outside the capital, so she never moves away. By that point, Azula is very much consumed with her duties as heir to throne and isn’t spending as much time with her friends as she used to, but with Mai still around Ty Lee doesn’t feel the desire to pull the trigger and run off and join the circus. Azula never has her friends “abandon” her. So the three Dangerous Ladies are together during a particularly difficult time in their lives, with Azula being trapped alone in an abusive household with the pressure turning up all the time and Mai and Ty Lee also having absolutely awful family lives where they’re getting badly neglected. And at critical moments they start leaning on each other(potentially in some pretty unhealthy ways) and open up about their issues, leading to them forming very strong bonds. This doesn’t mean that they understand how abusive their situations are(because awful family dynamics are normalized for them).
And the thing is that their friendship can still be very unhealthy in some ways. Ty Lee can still be leaning on her friends, particularly Azula, for the positive attention and approval she doesn’t get elsewhere in her life. Ty Lee can still prize her relationship with Azula as something which makes Ty Lee be unique among her siblings. Mai can still rely on Azula to inject her life with excitement and freedom from her family’s suffocating expectations. Azula can still rely on her friends for the love, affection, and support she should be getting from parental figures. There’s still a serious power discrepancy in the friendship which is unhealthy. Azula can even still order her friends around in combat situations, something that Mai resents. Fundamentally, the Dangerous Ladies are still three teenagers raised in an awful home environment at the top of a toxic, imperial system who end up getting thrust into a war zone to act as child soldiers, and that influences their friendship dynamics. At some level, Azula can still be terrified that her friends will abandon her or be forced to abandon her.
The other thing is that having a very strong friendship between the three of them does not automatically turn any of them, particularly Azula, into better people. Mai and Ty Lee are raised at the top of Fire Nation society and indoctrinated into Fire imperialist ideology, so, just like in canon, they don’t see anything wrong with the majority of Azula’s actions. Violently imposing Fire imperialism is a good thing for them, and they have little issue with most of Ozai’s actions. So, while Azula would absolutely listen to her friends if they actively tried to lead her toward a better path, they aren’t trying. If the Dangerous Ladies come to reject Ozai or Fire imperialism, it will be because their experiences cause them to or because events force them to.
Anyways, in Book 1 nothing really changes. In Book 2, there could be major changes due to the butterfly effect, but there doesn’t have to be. Maybe Azula recruits her friends when she is first given the mission to kill or capture Zuko and Iroh, and she still fails to capture her brother and uncle at the resort. Maybe, like canon, she doesn’t recruit her friends (via letter?) until she failed the first time. In either case, Mai and Ty Lee both come 100% voluntarily. Mai, like canon, is bored with her life and sees this as a way to get some excitement, while Ty Lee, without having run away to the circus, has nothing better to do and sees it as a chance to have some fun and help her friends. Besides, with both Mai and Azula out of town, she’d be alone with her “awful” family.
There really doesn’t have to be much change in Book 2, with the caveat that just about nothing which happens in “Return to Omashu” happens. You can still leave the other big Dangerous Ladies moments(“The Chase,” capturing Suki, taking Ba Sing Se) intact if you want to. Azula being a much better friend doesn’t necessarily change much.
Where things potentially change a lot are in Book 3. There are a lot of directions it could go there. Zuko could have a different dynamic with each of the Dangerous Ladies if/when he comes back, which could lead to different actions on his part. On the other hand, if you wanted, not much could change until the Boiling Rock, where things potentially go very differently. Azula could end having to choose, with her friends and her brother on one side and her father and her nation on the other. Also, the Dangerous Ladies could confront Zuko together, rather than Mai talking to him alone, potentially leading to interesting results.
Anyways, these are my thoughts on this AU idea. Again, I think it would be interesting to explore and I’ve never seen it done.