Lauriam and Strelitzia were
not in the same union.
First, consider personalities of Lauriam and Elrena. He is one of the top wielders of his union, while she is a bully. It seems obvious to me that these two would have known each other quite well if they were in the same union. Likely, some of the less strong wielders would have asked Lauriam to protect them from Elrena's bullying. But when they met after the war, it is clear that they didn't know each other beforehand. So, they must have been in different unions. Strelitzia on the other hand was in the same union as Elrena, since they were doing missions together.
Second, why did Lauriam join the Dandelions? While he might not like the idea of a war, he would still have seen the it as an opportunity for himself. If Strelitzia is in the same union, that is also a reason to stay in, to protect her. But if they were in different unions, then there is a risk that they end up having to fight against each other, and that gives him a good reason to join.
Then, a few other observations:
- Ava and Brain knew each other very well from the start, so Brain was obviously in Vulpes.
- My headcanon puts Ven in Unicornis. I don't think the keyblade would have chosen Ven as a wielder on its own, at least not at such a young age. The Master of Masters probably made it happen, because Ven has an important role in his plan. Then he would have assigned one of the Foretellers to take care on Ven until the war happens. The ones best suited for this would be Ava and Ira. But we know it wasn't Ava, because Ven himself said that he didn't know her.
- Lauriam's attitude, especially in the Siblings scene, is very similar to what Aced would teach to the members of his union.
- I don't think that Ephemer and Skuld were in Vulpes. I see that majority of the regular Dandelions would belong in one of two big groups. First, wielders from Ava's own union would have found it much easier to join than others, because they didn't need to worry about betraying their old leader. Second, there is the group recruited by Skuld, who are mostly from the same union she was in. If that is also Vulpes, then the Dandelions are essentially Vulpes v2.0, which is surely not what the Master of Masters intended.
And before anyone else points it out, I'm aware that I'm probably going into way more detail here than what Nomura himself has ever thought about this.