I wouldn't do it, the chances of the older chicks cannibalizing the lone chick is a real possibility. I accidentally made that mistake once. So I'm fore warning your thoughts on this.
With a broody hen looking after the single chick, I would be more afraid of the broody killing the month-old chicks that are not hers. But of course either way is possible, depending on the individual broody and how she behaves.
This is one of those cases where a lot will depend on the temperament of the specific chickens involved, especially the broody hen. Some broodies will try to clear all other chickens out of a very large space (I've seen some that wanted 10+ feet in all directions, which is impossible in many sizes of coop & run.) Some other broodies will tolerate anyone else quite close, and may not protect their own chicks well enough. I've also read of broodies that try to adopt all chicks of all ages, who might happily take in the month-old chicks as well. I don't know about your particular broody, but you will probably just have to try something and watch, being ready to intervene.So I currently have 5 3.5 week old chicks in a brooder....I have broody and her chick out in a sectioned off area within the coop. It has an area under the poop board that is inside the coop, and a seperate area within the covered run. I’d like to move my chicks out in a week or two into that space so that they can begin integrating with the flock.
Think it’s likely that they can share space peacefully with broody and her baby?
Personally, I might try letting the broody and her chick mingle with the rest of the flock, while watching carefully to see what happens. If that works, you could use the sectioned-off area for the month-old chicks. You would need to watch if the broody and/or her chick are in danger, but also make sure she isn't bullying the other hens too much.
Making another sectioned-off area would probably be safest for the various groups of chickens involved, but I don't know how practical that is.