You have more room than I expected. To convert that big one is roughly 8' x 13' and the smaller about 5' x 10'. That's runs, not coops, but it isn't bad for nine chickens. Plus there is some possibility of free ranging.
I'm always in favor of more room. If you can I'd move the two together but keep a gate or door between the two where you can let them mingle or isolate if you need to. That gives you a lot more flexibility now and in the future.
I don't know how big of a threat that cat really is to those chicks. Many people keep barnyard cats around their flocks with no issues, even with a broody and baby chicks. Others can have real problems. I can't assess that from here.
I'd think the fox and badger would be as big a threat to your adult hens as they would be to those 6 week old chicks, but there other predators, smaller hawks for example, that would take a chick while passing on an adult. I would not think the chicks are at that high a risk free ranging but they are at more of a risk than adults. The way I'd handle integration would be to let both groups out to free range together but go back to their own separate coops at night. I can see why you might not like that suggestion though.
That brute of a hen bothers me. Normally hens are not that aggressive toward chicks. They may initially be curious but typically if the chicks don't bother them the hen doesn't bother the chicks. Sounds like you may have a psychopath. Also most broody hens are so protective any other hen doesn't have a chance. Sounds like you got unlucky on both counts. In many circumstances I'd agree to let them fight it out but between a 6 week old chick and an adult hen it is not a fight, it is a massacre. It doesn't sound like that broody would be much help to the chicks.
Sometimes when managing chickens you have to make hard choices, like you are talking about with the cockerels. When I make those decisions I try to think about the best for the flock as a whole, not for any individual chicken. It may be time for that older hen to go.
Are both those older hens laying? If they are I'd be reluctant to teach them to lay anywhere other than the nests. That may be another constraint on you. With your constraints on free ranging the chicks I'd try putting them all together in that big run when you can be around to observe and see if you need to do anything with that older hen. It may work out OK. If not, you can try isolating that hen, maybe in one of those smaller coops, for a week or so before trying again. But you may be faced with the decision between her and the four pullets. It's not always easy.
