So your coop is about 4' x 6', your run is about 6' x 8' and you have 10 hens. Building an additional section of run will certainly help, I'm surprised you aren't having behavioral problems with what you have now. And you want to add more chickens to that. I consider that very risky,
If you lock the broody hen in the nest she needs room to get off the nest and poop, doesn't sound like she has that. She will hod it as long as she can, but if she poops on the eggs they will almost certainly get bacteria inside them and kill the chicks long before they hatch. You might be OK giving her two of those nests and tearing down the divider, but leave a lip at the bottom to keep her nest on one side. Doing that does not sound practical. Some people build a pen around the nest to give her room to poop but your coop is too small to do that inside. With the nests hanging off the side that does not sound easy either.
One option might be to open up the nest once a day, take her out, and set her on the ground. She should take off after a few minutes, eating, drinking, and taking a poop. At some point she will go back to that nest, but it could be 15 minutes or over an hour.
If you are going to leave her in that nest I would not lock her in. Let her come and go as she wishes. Mark the eggs so you know which ones belong (I use a Sharpie) and check under her every day after the others have laid to remove any that don't belong. Anything else just doesn't sound practical.
In your situation I'd try to move her. I consider your coop/run set-up too small. Set up a nest in that brooder box, it should be big enough. There is a chance she will break if you move her but many of us do it all the time. Many committed broody hens do not break. I'd make the nest itself pretty dark, that seems to help. I'd also make the nest so you can lock her in the nest only with that making the nest really dark. Let her have her freedom in that other nest for the day before the move so she poops and such. Then, after it is really dark, move her and the eggs to that nest and lock her in. Leave her locked in total darkness the nest day until really late, just a little daylight left. Then open that door so she can get off the nest and leave it off. She might come out then but probably not. She will probably come off the next morning to eat, drink, and poop. I can't give you guarantees but this almost always works. If she absolutely refuses to accept the move and relentlessly paces that brooder for an hour or so, let her out. She should go back to her nest. Give her the eggs, they should be OK.
So what do you do after she hatches, whether in that brooder or in her original nest? She might be able to take care of them with the flock, especially with extra run space. With them shoehorned in there that tightly it will be harder for hr to take care of them, but most of the time my other chickens aren't that much of a threat. Of course I have a lot more room. If they could all free range all day it would make it a lot easier.
But at some point she will wean them. I've had some wean their chicks at three week, some go almost three months before they leave hem to make their way with the flock. That is the time I'd really worry about. Even my chicks that were three weeks old when weaned were able to take care of themselves. One way they do that is to avoid the adults. They form a separate sub-flock and stay away from the adults during he day and avoid them when sleeping at night. Even with the additional run space and especially with that tiny coop you don't have room unless you let hem all free range all day every day, but that still leaves that tiny coop.. I think with that limited run space and especially that tiny coop you are headed for disaster, whether you let her try to integrate them into the flock for you or even if you keep them separate until they are fully grown and try to integrate them. You do not have the room it is going to take.