Unfortunately, I don't own a hardee. However, from a non-professional standpoint, I'd think sewing it is a good idea. If you ever want to remove it for any reason, you can simply cut the thread. There is the problem with glue becoming a permanate fixture if you decide to go that route.
As far as sewing it on, I'm sure there is some way to get a small and very fine needle through the feather and hide the visible stitches in the hat cord if you "sew" desire.
Just an idea. I'm sure there are other professionals here who can help more.