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.
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.
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.
The Hashiras would likely show concern. Some, like Shinobu, would immediately start looking into possible medical solutions. Giyuu might be more reserved in his worry but would still keep a close eye on Tanjiro. The other demon slayers would also be affected. They look up to Tanjiro, so his sickness would lower the morale of the group. They would try to support each other while waiting for Tanjiro to recover.
Well, some characters might be amused. For example, Atsushi could be a bit shocked at first, seeing the different interpretations of Dazai in the fanfiction. He might find some of the more out - there scenarios quite strange but also interesting in a way.