The texture rect idea you had before was a pretty good solution. Have each hair of the cross hair be a different TextureRect and manipulate it with a AnimationPlayer.
center:
left:
right:
up:
down:
you would have to create a texture with smoothing at the edge of the hair and set the texture filter mode to linear for it to work
also the pictures I provided might be too low resolution to create the effect you desire so you would have to create a texture with a higher resolution to be more smooth