Though some of the Batwoman characters have played games for survival in the past — like Sophie, Ryan, and Mary taking on Cluemaster — it's not a typical crime-fighting night for the team. Some of them would certainly be better at playing children's games in a Squid Game-style event than others.
Of course, winning Squid Game isn't just about playing the games in each round. Competitors also have to survive any conflict in the dorms, stay off of the guards' radar, and be willing to make and betray allies along the way. Not every Batwoman character would be able to make it to the end.
Mouse
If Mouse somehow managed to make it beyond the first few games of Squid Game, he would have a hard time going any farther. After all, Mouse has deferred to Alice's decision-making skills since he was a child. He isn't exactly a leader or someone who makes his own plans.
That would make the later rounds, like him having to betray a partner in "Marbles" or figure out a safe path in "Glass Hopscotch" impossible. He wouldn't survive the games.
Marquis
Marquis is one of the few Batwoman characters who would likely love the idea of Squid Game. He would be disappointed he hadn't thought of it first. Taking advantage of people and putting them in painful situations would certainly appeal to him.
Marquis wouldn't be able to survive the games himself, though. The other players would certainly be quick to see how much he enjoyed the danger, and they wouldn't want to ally with him. He'd be someone most of the other players would freeze out, or end up targeting because they didn't want to face him later.
Kate
Kate Kane might not have any superpowers, but she's certainly an incredibly observant character who would want to figure out just how Squid Game was developed, who was behind it, and how she could put a stop to it. That would all make her a target of those running the games.
As the series demonstrated, getting in the way of the game's progress could make a player a target of those behind the scenes, leading to "accidental" deaths or pitting other players against them.
Luke
It's hard to imagine Luke would allow himself to even be lured into Squid Game by a stranger promising him thousands of dollars. His competitive nature might get the better of him, though, and if it did, he'd be in trouble.
Though Luke can hold his own in a fight, his best skills are based on his manipulation of technology, and Squid Game wouldn't allow him to showcase that. While he could contribute and help his allies win in "Tug of War," Luke's tendency to think he always knows best might rub some of the other competitors the wrong way. He would definitely find himself betrayed by an ally in the later rounds.
Ocean
Ocean is no stranger to having patience or fighting for his life, so Squid Game wouldn't be out of the realm of possibility for him. Where he might find himself falling short is in bonding with other competitors.
While Ocean quickly develops a rapport with Beth Kane, he's the only person the audience sees that from. Otherwise, he keeps his emotions to himself and is a hard person to get to know. If other competitors didn't think they could trust him, they would only see him as a threat.
Alice
Alice would probably be very amused by the concept of the games. She would enjoy them at first when her particular brand of creativity would allow her to skate by the first few rounds. Where Alice would falter would be in the final rounds as she realized that no one trusted her and would be expecting her to betray them.
It's hard to say whether Alice would meet her end by being bested by someone else in a round of "Marbles," or if she would try to appeal to the game master directly and end up removed from gameplay. Alice is a bit of a wild card, and her path could take her out of the games or right to the end.
Sophie
With her military and security background, Sophie wouldn't be the kind of Squid Game competitor who could just ignore how the unjust nature of the games. She'd probably spend just as much time trying to investigate the games as she would competing in them, only trying to stay in to bring the whole place down.
It would be her mistake to think that those behind the games wouldn't know why she was really there, however, and she would be found out and eliminated as a result.
Mary
Mary would initially seem like an underdog in Squid Game. She knows how to cultivate her image, and she did it very well for years before revealing her intelligence and underground clinic to her family. She's one of the most likable Batwoman characters, and as a result, she would be able to blend in with the pack well during the early rounds.
Mary would also have the patience and the steady hands of a surgeon, for a game like the "Sugar Honeycombs." She stays calm under pressure, something she's demonstrated with her dealings with the new Mad Hatter. What it would really come down to is whether or not Mary would be able to fight someone one-on-one.
Julia
Julia Pennyworth has worked in security and for some mysterious government organizations in her day. She's the closest thing the show has to a spy. As someone who regularly puts her life on the line as part of her job, and then does it again when she pretends to be Batwoman for Kate, Julia is easily one of the bravest Batwoman characters.
She would likely be able to get through the physical challenges and the mental games of Squid Game pretty easily, making and breaking alliances as needed. Where Julia might falter is in the end as she would be willing to sacrifice herself if she thought one of her final opponents should be the winner instead. That's something she would have in common with Luke, Sophie, and Kate if they made it that far in the games.
Ryan
There's no denying that Ryan Wilder is a survivor. She's managed to make it through threat after threat to her life and the lives of her friends at every turn. She would hate to have to engage in Squid Game, and she probably would try to blow it off initially, but once she realized that the games weren't a one-time thing and that people were repeatedly being put in mortal danger, she would be all in.
Ryan would hate that so many people would die in order for her to win, but like Mary, she would do her best to blend in and figure out a plan before going all out to make it to the end. Unlike Mary, Ryan has the physical abilities to be able to battle a competitor in a one-on-one fight as well, making her the one most likely to come out on top.