This is me talking myself out of using wavey braid in a straw brim.
Where it is seen:
- Decorative insertion work on undergarments. White cotton.
- Decorative insertion on straw plait baskets. Color cotton.
Where it is Not seen:
- On dresses or outerwear garments
What is seen inserted in brims:
- Other color straw plaits
- Other natural fiber braids – other grass-like fibers, wool braids in single and multiple colors
- Horsehair braids
- Bias cut silk (textual documentation)
If I Had My Own Blue Box: 
Love the wavy braid in the basket!
unless you can provide period documentation for the use of wavy braid on a hat, I think you might regret using it. I know you pride yourself on your accuracy! It is pretty though