My cat does the same: she wakes me up at 4:30 or 5:30 every single day, except on weekends.
I have tried all the things you suggest: I have another cat (her sister) but this one sleeps or looks out the window and does not wake me up, which is great, but they don't play together or entertain each other.
I have also tried to keep her out of the room, but that's even worse.
I even open the door for her so she can go out to the garden, which calms her for about 10 minutes.
When I get home in the evenings, I play with them and brush them and they are stuck to me until we go to bed, but it doesn't make any difference.
Not only that, but she screams (her meowing cannot be described any differently, as she is very loud and really sounds like a baby crying), so my neighbours are not very happy.
Any suggestions are welcome to keep me from going crazy.