If the cockerel is the instigator it will certainly help. He very well could be. If you can try isolating him for a few days you can see what effect getting rid of him would have.Are you thinking my original pullets will calm down about the new girl if I get rid of the cockerel?
Sometimes the pullets stay in the coop and up on the roost to avoid the cockerel when he is overactive in puberty and mating them a lot. That's even when they are all the same age and grew up together. I'm not looking at them but that does not seem to be what is going on here. One or more of yours are attacking her so she is avoiding them. I don't know which one.
With my flock it is pretty common for my younger chickens to avoid the older. The more mature outrank the less mature and are often not shy about enforcing those pecking order rights by pecking (or more) when the younger invades their personal space. It usually doesn't take long for the younger to learn to avoid the older. Mine typically don't go out of their way to attack the younger as long as they keep their distance. Every now and then you can get one that will be a brute, that can be male or female. Each chicken has its own personality.
Sometimes mine will merge earlier but typically they form sub-flocks until the pullets mature enough to make their way into the pecking order. That is typically about when they start to lay. Size doesn't matter, it's laying eggs that seem to admit them into adulthood. That's when they start roosting together. eating together, and hanging together during the day.
Having a single chicken makes it worse. They are flock animals and want to be with other chickens. But if they invade personal space they can get pecked. It's sometimes worse if the single chicken is the older one. They want to be with the other younger chickens but will still peck if they get close enough for personal space to be violated.
You will have great space once they are all integrated but integration takes more space than just living together. Your pullet should have enough to avoid the others but she may have to use the room in the coop to achieve that.