My brooder is in the coop. I put chicks in there straight from the incubator or post office, even if the outdoor temperatures are below freezing. As long as one area is warm enough in the coolest temperatures and one area is cool enough in the warmest conditions they do a really good job or regulating their own temperatures. To me the biggest challenge to brooding out of doors is the wild temperature swings. I've gone from the 20's to the 70's in 36 hours. The brooder has to be able to handle both.
I have a lot of area outside fenced in. A 12' x 32' main run and an additional 45' x 60' in electric netting. My adults spend practically all day every day outside except for when laying eggs. Since mine are essentially raised with the flock my integration consists of opening the brooder door when they are 5 weeks old. That's it, I just walk away. It can be that easy.
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Even if you were willing to brood in the coop I'm not sure you have time to set it up. So what I suggest is that you build a "grow-out" pen in that big coop you have. Or finish off that other section where you are planning on building that coop. I don't know what you have for a run, but either section off part of your run or build a new one that shares a fence. Make a way to let the chicks out into their section of the run. After they have been across the fence from each other for a few weeks, try integrating them.
Will this work each and every time? It does for me but it's always possibly you have one of those hens that is a true brute. Or you may not have much room outside. If they are shoehorned together there is a lot more risk. They need room and I don't know how much you have in a run. Occasionally free ranging them doesn't count.