Most people just straight out put the cat in the same space as the others (so right out of the bat, you're doing it better). Our cats are exceedingly tame and are used to cats coming and going, so we've not had this behaviour, but it seems very normal to me. Cats identify each other by scent; until they get used to the new cat's scent, they're going to think "there's an intruder in the colony"; and if each other is the only cat they can see, it does make some sense they believe the scent belongs to the other. It's really weird for us humans to think of seeing someone and not quite recognizing them, but again, cats identify by scent, not sight.
My guess is that it should be perfectly OK. In a week or two they should have grown accustomed to the smell, at which point they should have stopped growling at each other (probable much sooner) and you can let Rayla out of the room.
Awesome! Thanks for the info, that's a huge relief.
It's my first time adapting a new cat, so it's been a wild ride!