You DO want the saddle girthed up, but not tight as if you were going to ride. Just secure enough to keep the saddle on.
When you ask her to canter online, do you do it with her saddle on?
Does the bit have a single joint?
It may be that she doesn't like the nutcracker action (seen in single jointed bits)
My mare used to contanstly flip her head, until I got her a bit that was double jointed. It stopped it instantly.
So that's something else to look into.