It's not possible because meshes are made up of polygons and FFD amounts to moving the points of those polygons around. This builds on what 3D graphics has been able to do for over a decade and how graphics processors today are built (they're optimized to handle vertices and triangles and portions of images placed in those triangles).

My guess:
Bezier curve warps (in real time) would require a totally different technology or require computing power that most GPUs simply don't have.
Even rendering stuff like that in AfterEffects or some other program takes a significant fraction of a second at least to do.

Either way, it's totally different from how FFD and game graphics in general does things. Doing Bezier warp stuff would involve making it from scratch and, I think, is kinda outside the scope of what Spine is designed to do.
It may become possible/the norm someday though. There was a time after all when complex 2D skeletal animation was too computationally expensive to use too much in games.

Nate did mention that FFD was only the foundation of more powerful tools later.
What you described is possibly one of them. Certainly makes sense to me.

Though I think the Skinning feature Nate's currently working on will possibly bend meshes into curves too. It'll be a different way of doing it but you'll get the results.

Like Pharan says there is no real simple way to do this, but you will be able to use skinning to get the result you are after. Both FFD and skinning is something we will be expanding greatly upon.