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.
Another theme is redemption. Fanfic authors like to explore the idea of Akutagawa making up for his past wrongs. He might be given a chance to save someone or change an event that he regrets in his past. It gives a new dimension to his character.
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.
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.
One great BSD Akutagawa crossover fanfiction is 'Akutagawa in Another World' which explores what would happen if Akutagawa was transported to a different universe. It has an interesting plot where he has to adapt to new powers and a new society.
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.