Definitely not. Batman has a large and diverse fan base, including many young fans. Such 'fanfic' with adult - related and improper content has no place in a fandom that should be centered around positive aspects like Batman's fight against crime and his moral code.
It can divide the fandom. Some fans might be extremely offended by it and distance themselves from those who support or create such fanfics. It could also create a negative image of the fandom in general, making outsiders think that the Batman fan community tolerates or even promotes such inappropriate content.
Fanfic in general helps keep the fandom alive and active. Batman and robin slash fanfic specifically can attract different types of fans. Some might be interested in the relationship aspect, while others might be drawn to the unique twists on the characters and their adventures. This diversity in fan interests helps the Batman fandom stay vibrant and evolving.
Batman would probably isolate himself at first. He'd retreat to the Batcave to analyze the situation. He might feel a sense of guilt, thinking he should have seen the betrayal coming. After that, he'd start planning his counter - move. He'd use his vast resources and knowledge to expose the truth behind the betrayal and take down whoever is behind it.
It adds a new dimension to the relationships within the Batman world. Fans get to see a different side of Batman through his interaction with Black Canary. It can also inspire new fan art and discussions.
It gives fans a chance to explore different 'what - if' scenarios. By seeing the Avengers' reactions, fans can get a deeper understanding of the characters.
He might isolate himself from the League at first. Batman is used to working alone, so he could retreat to his Batcave and plan his next move. He could also start to prepare counter - measures in case the situation escalates and the Justice League becomes a threat to him.
Batman would be heartbroken. He has trained Robin and sees him as a partner and almost like a son. He would first try to talk to Robin, to find out what made him turn evil. He would hope that he could reason with Robin and bring him back to the good side.