The Flash might initially be in denial. He's used to Batman always having a plan and his death would be a huge shock. He would then turn his speed into a tool for investigation, trying to find out exactly what happened. Green Lantern could feel a sense of loss but also be spurred into action, using his ring to try and prevent future tragedies like Batman's death. He might create constructs in Batman's memory to inspire the team.
One possible storyline could be that Batman sacrifices himself to save the world from a powerful alien threat. He uses his intellect to devise a plan that requires him to give his life. Another could be that he is poisoned by a new villain in the Justice League Unlimited universe, and despite the efforts of his friends, he succumbs to the poison.
In a fanfiction like this, the gun could be a symbol of the danger that the Justice League, especially Batman, constantly face. Deadman, being a supernatural element, might be trying to communicate something about the gun's origin or the person wielding it. Batman, with his detective skills, would start investigating while the rest of the Justice League deals with the immediate threat of the gun - related situation.
In a fanfic where Batman is injured, it could lead to many interesting plot developments. Maybe the other Justice League members have to step up their game. Superman might take on a more protective role over Gotham in Batman's absence. Wonder Woman could be the one to lead the investigations that Batman usually handles. This kind of injury to Batman can create a shift in the power dynamics within the Justice League.
In many Justice League fanfictions where Batman dies, it could lead to a complete shift in the team dynamics. For example, the League might become more reckless without Batman's strategic mind. Superman could struggle with the loss, perhaps blaming himself for not being able to protect Batman. The other members like Wonder Woman would have to step up in terms of leadership and planning. It could also open the door for new characters to emerge in the fanfiction's storyline, trying to fill the void left by Batman.
I really like 'Batman: Shadows in the League'. In this fanfic, Batman has to deal with a secret threat that is undermining the Justice League from within. It shows his detective skills as he uncovers the mystery. The characterizations are on point, especially how Batman interacts with Superman in a more tense situation than usual.
Well, it often makes the team more protective. They start to realize just how vulnerable Batman is despite his skills. For example, Superman might become more vigilant about looking out for Batman during battles.
In some fanfictions, Batman might be injured while investigating a dangerous conspiracy. He gets too close to the truth and the people behind it try to eliminate him. They set up traps, like a high - tech booby - trapped building. Batman is caught in an explosion or a collapsing structure, which results in his injury. Then the Justice League has to come to his rescue and deal with the conspiracy.
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.
Yes, there are likely some out there. You can check fanfiction websites like Archive of Our Own or FanFiction.net. Just search for 'Justice League Batman Gun Deadman' and you might find some interesting stories.