I think it must have been before the whole business with Luten. It might have been something he didn't think about for years. For example when he faced the dragons. I had a theory that he might have found out where they were from The White Lotus, much like the Masons and the Knights Templar and even the real White Lotus had esoteric and secret knowledge of the hidden mysteries of the world.
Even though he was obsessed with his Pai Sho tile, I think he was reluctant to reach out to them until Zuko was safe.
I would argue that he was only Loyal to Zuko, and had went along with him so no one would murder the kid. If you were a cynical person, the obvious solution is that once Zuko's life was threatened he had to run, and that the White Lotus could have kidnapped Zuko and thus adopted him and then trained him to be a good ruler and defy his father. A normal person would probably want vengeance at being hurt and humiliated. But Zuko was so damaged and suffering so much emotional trauma he didn't want to leave his abusers.
They might have pressured Iroh to kidnap Zuko, instead of just leaving him to die in the Wilderness on his own. Which seems odd, no one wanted Zuko around in the Fire Nation. but it would have been a nonviolent solution to undermine the rule of the Fire Nation, if their had been no Avatar. It was probably the only option Civil War and a coup detat in the fire nation, between Azula and Zuko.
But more likely Iroh decided that the pressure would be too much on Zuko and it was important for him to find a safe place to recover from his emotional abuse. When none could be found they went on the run. The only place Zuko would ever be safe was with Iroh. He commented that he wasn't making a new life in Ba Sing Se. Its not that Iroh was going to let his Nephew become a coffee shop waiter, if he wanted more out of life. He just knew that he needed to heal in a safe low pressure environment. Unfortunately Iroh couldn't save Zuko from himself and Zuko scurried back home like a beat dog. It couldn't last, and Zuko having known real love and a safe violence free environment had healed enough to know that this was not the way he wanted to live.
If there had been no Avatar, or Iroh and Zuko had never teamed up, I think that Zuko might have been the one to crack and go crazy. Probably even worse than Azula. Even if the White Lotus used him as a pawn he would have been in poor mental shape.
Getting into fights with the Avatar and getting exiled to Ba Sing Se probably saved his sanity, as even though they seem pointless at that point they actually gave his life meaning. A few weeks in Ba Sing Se, without risk of being killed or beaten or emotionally annihilated, might have been as close to a normal happy life he had ever lived. And that is really sad.
The White Lotus might have used their resources to help put Zuko on the throne if Ozai ever died... not knowing he was crazy too.
i wish we knew more about this group.