I looked up a tutorial for even count flat round
peyote stitch, but no matter how carefully I follow it, I end up with
a round piece of peyote that's ruffled rather than flat.

It sounds like you're increasing too much. Nothing else I can think of would
make it ruffle except that. Cut back to one bead increase instead of two, or
to every third row instead of every second and see how that works for you.
HTH

Cheri
(Bubbee to Emily and Nathan)

It's my life
And it's now or never
I ain't gonna live forever
I just want to live while I'm alive - Bon Jovi