They won't move into the bigger coop on their own. Right now I'd let them spend nights wherever you currently have them, and only day time with the older birds. After 1-2 weeks, if all goes well during daylight hours, I'd look to move them into the coop at that point.But what about night time? I just got the chicks used to sleeping in their own coop at night and I don’t know if they will go into the bigger coop on their own.
My other option is to leave them in the smaller coop until they are 8 weeks old and then put them on the perch of the big coop some evening and not create the gate at all.
I would not do option 2. That gives them zero chance to interact with supervision, and it's a crapshoot what you'll open the door to find the next morning.