I agree, I'd wait on the shanked bit, especially since she is apparently not even comfortable with a bit.
I'd try a snaffle like a French Link, or one with copper rollers in it.
If she doesn't stop, You can always try using a cavasson on her for awhile too, and see if it's just a habit. I've had horses who would play with the bit, just to play with it, but after a few rides with a cavasson, they stopped. Put the cavasson on loosely enough that you can get 2-3 fingers under it, that way she still can move her jaw when she needs to.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."