No - you don’t have to keep them in a brooder for 6 weeks to keep them safe from a predator. My 3 week old chicks have the run of the 8x8’ coop, and will soon be in the run as soon as I get the fence reinforce. As long as the coop and run are predator proof, they are fine.
My chicks have been out in the coop since I got them at 2-3 days old (however long it takes to ship them). They are currently 2.5 weeks old. They have the MHP brooder system and are weaning themselves off of it, even with a window taken out of the coop. (It’s supposed to start raining this week...
Sounds like you need some more room before you try integrating anything again. It appears that the submissive ones weren't able to get away from the others. It will help if you can also put some hiding places in the run for the others. Pallets or plywood against the fence (but open on both ends...