For the best results waiting until they are 16 weeks old is optimal.
However, I have gotten away with integration of the youngsters earlier than that. I would say that the longer you wait the better, generally speaking.
Since your primary flock consists of two hens instead of 50, if it were me I'd give it a trial where I sit out there for TWO hours with them and see how it goes with integration. If you can wait another couple of weeks the chances get even better that things will go well.
Everything depends on the personalities of the two barred rocks. If they are not too aggressive things may go well. For example I have placed adult silkies in with chicks younger than yours and all was well.
If you see the younger chickens with tushies in the air against the wall or fence, and the older hens attacking from the rear, get them back in their crate!!!! They are probably about to be killed.
If you see the barred rocks pecking viciously at the youngsters but only once or twice, with the youngsters running away safely, things are going well.
I LOVE Golden Campines. I used to have 20 of them and they are pretty active birds. That is in their favor here. I have never had Barred Rocks and so cannot speak to their disposition. If you had Rhode Island Reds or another aggressive breed I'd wait longer.
Also- if your run is very small I WOULD wait longer. If there are no places for the youngsters to "get away" as in across the yard or in a large run, I think I would try to hold off for just a bit.
There are no guarantees and it is always a risk that the chicks will be hurt and especially before 16 weeks or they are about the same size as the adults. The key is to set aside a couple hours that you can sit there in the lawn chair and watch for trouble when you do it. Then if things are dangerous you can save them. Some integrate at night but I like to see what is happening.