That's an adorable hat!
I only have 2 ideas. One, when you're done the hat, block the brim. That hopefully will keep it flat.
Or, if you'd rather go back or start over, where you have 2 rounds of seed stich, that may not be enough to keep it from curling. Maybe do at lest 4 or even 5 or 6 rounds of seed sitch before starting the stockinette stitch.
Good luck - let us knkow what you end up doing.