A cursory glance at the internet shows that perhaps the closest (if not exactly) to what I'm seeking is Albert Nijenhuis' generalization of the ordinary Lie derivative. He constructed a way to take the derivative of a differential form along sections of the bundle of differential forms taking values in the tangent bundle.
Does anybody else know any other interesting formulations of the Lie derivative close to what I'm seeking? I am curious about applying the Hamiltonian theory of elementary symplectic geometry to this setting.