After doing some research you might be correct about it being safe for him, as people who go in can't age or die, but I still feel like a conventional cell would be less intense and more fitting for a person who, as far as the league knows at this point, is only a crazy stalker. On another unrelated note, some of the league members don't have or have a need for a secret identity, such as wonder woman, who doesn't need to protect her friends because they have superpowers anyways.