Afraid I’ll have to disagree with that one. One way chickens have worked out to live together is that when there is a conflict, the weaker runs away from the stronger or just avoids then to start with. One of the worst places I’ve seen for brutality during integration is on the roosts when they are settling in for the night unless they have plenty of room to get away from the bully. Those high roosts allow them to get away and avoid, day and night.
By providing separate food and water, you help them avoid confrontation. It’s a common tactic for the stronger to intimidate the weaker to maintain pecking order status. That’s part of what is going on. They are determining the pecking order.
It sounds like something else is going on too. Chickens can be territorial. They know who belongs in their flock and who doesn’t. It doesn’t happen all the time but occasionally a chicken will try to run off or attack a strange chicken. Sometimes they welcome them into the flock. They are not consistent about this.
There are a lot of things they are not consistent about. There is a difference in what might possibly happen and what definitely will happen 100% of each and every time. Stopping egg laying when there is a change, for example. Some do, some don’t.
They will eventually work out the integration issues. How long will that take? Who knows? If you have a large group of people in a room, why are some too warm, some too cool, and some just right? They are all different. It may take days, it may take weeks; they are all different. But they will work it out.
It won’t hurt a thing to put a nest up there if you can as long as they don’t start sleeping in it. That’s the risk, so make it lower than that high roost. You can leave it up there as long as it doesn’t cause a roosting problem or remove it when they work it out. Just go with the flow.
Don’t worry about that second soft-shelled egg. In all the excitement and stress, her internal egg laying machine messed up and released a second yolk before it should have. If they release two yolks at the same time when they mess up like that, you get a double yolked egg. When a second yolk is released a little later, the shell gland often does not have enough shell material to cover a second egg so it is very thin-shelled or even shell-less. If she makes a habit of it after they settle down you may have a hen with a genetic defect, but right now I would not worry about it.