Yes to caging the bully. But I meant for nighttime since you specifically said the issue was cooping them together at night. I'd want the new birds (and the existing hen that's not causing issues) to have the opportunity to learn to get along in the coop, without having a bully wrecking things.
Actually it's not, lower ranked birds are far more likely to bully young newcomers because they don't want to lose more ranking.
Going to be tougher to integrate with the eglu because realistically there's no space to add barriers to separate birds in one, versus a more traditional coop and...