This sounds like one side of the general dilemma of inheritance in OOP languages: Actual inheritance lets you get the base functionality "for free" but lets you potentially break the base class contract (which is why sf::Sprite::draw() is and should be private), while not using inheritance means you have to write all the tedious wrapper functions.
In this case the sf::Sprite interface is not very big, so I'd say do the wrappers. Maybe you could even use an (evil) macro to make the wrapping process less tedious and verbose.
As an alternative, you could do composition with a sprite and then simply make the sprite public so that you don't need to do all the wrappers.
As another alternative, imo "animation updates" belong in an update() method, separate from the draw() method, so you could simply inherit from sprite, add an update() method, and leave it at that.