notsure if that is what you want but if you Export as alembic, and import again, it creates a mesh sequence cache modifier with the animation. or what else would you bake it to?
Use blend shapes. Add a blend key, apply the displace modifier, that should work. Other than that, blender does not export the information you're looking for. Although it might be possible with the new alembic support? I am not entirely sure. My recommendation would be to do it with blend shapes though, and actually hook…