There is some weird (for lack of a better word) way to do intarsia in the round, but I can’t recommend it because I know nothing about it. If I wanted to make this hat with an owl with a different color I would make the hat flat to do the intarsia. I think it will be a lot easier that way. It is not hard to change a basic hat pattern like this to flat. I think you want to have k2 to begin the hat and end it P2 with a seam st added on each end at the beginning and end. I’d make them both knits. You can follow the shaping as it is given just keep having that 1 extra stitch on each end for a seam.
The intarsia will work just as well changing from knit to purl and back as you will need to do for the owl. Just always make sure when you change colors that you have the yarns on the back side when you “twist” them and then bring the one you are needing to use next to front if you need to purl and to the back if you need to knit and it will work fine.
Seam the hat when you are done with mattress stitch and it will look very nice.