In the BSD manga, unfortunately, Dazai did die. His death was a crucial moment that influenced the direction of the story and the relationships among other characters.
One common theme is identity exploration. Since Dazai is a complex character, when he crosses over into different universes, he often has to re - evaluate who he is. Another theme is the clash of different worlds' values. For example, if he crosses into a more morally strict world, his rather ambiguous moral compass creates interesting conflicts.
Stray Kids, on the other hand, would also be very concerned. Bang Chan, as the leader, would take charge. He might start asking questions to figure out what exactly happened. Felix could be bringing water or something to make Y/N feel better. Lee Know would be standing guard, making sure no further harm comes to Y/N while they are in a vulnerable state.
One great 'bsd dazai crossover fanfiction' is 'Dazai in Another World'. It takes Dazai from BSD and places him in a fantasy setting. The story plays with his character traits in a new and interesting way, exploring how he'd interact with the new world's rules and characters.
Y/N should have their own goals and dreams. If they're interested in becoming an Auror, that can shape their actions and decisions. Their reactions to Draco should be based on their personality. If they're a shy person, they might be more hesitant around him. And make sure their dialogue is natural. Don't make them say things that are out of character just to move the plot along. For instance, if Y/N is a studious type, they're more likely to talk about books or spells when interacting with Draco.
Well, since it's related to Bungo Stray Dogs, perhaps it's about a special entrance exam within the agency or some other organization in the story. Dazai is a complex character, so it could involve his schemes or his interactions with other characters during this exam. It might also explore the rules and challenges of this exam, which could be unique to the world of BSD.