Man, fuck the "lore". If we're being realistic here Batman would have had eyes on Luthor from the second the game started. He would have noticed that Batgirl hadn't been on patrol in a long time, tracked her movements during the daytime, found Luthor's base, and the Justice League would pancake it. Any attempts to justify anything via lore is completely pointless as Batman, the Question, Superman, Green Arrow, literally any male hero with a connection to any of the controlled women would have figured something out, alerted the rest of the League and have Luthor in prison for kidnapping, rape, and a dozen other crimes inside a few days. Not doing anything because Batman would figure it out is silly since he'd figure everything out regardless.