We study L_\ell-biconservative timelike hypersurfaces in the Lorentz pseudo-sphere, defined by isometric immersion \xi: M^n_1\to\mathbb{S}_1^{n+1}. The operator L_\ell, defined in terms of the \ell-th Newton transformation, generalizes the Laplacianand arises as the linearized operator of the (\ell+1)-th mean curvature.A hypersurface is said to be L_\ell-biconservative if the tangentcomponent of L_\ell^{\,2} \xi vanishes.We prove that the $L_\ell$-biconservativity condition hypersurfacesin $\mathbb{S}_1^{n+1}$ forces the $(\ell+1)$-th mean curvature to be constant. The proof is carried out through a detailed analysis based on the four canonical forms of the shape operator ofLorentzian hypersurfaces. Examples illustrating the obtained results are also provided. This work extends the classical theory of biconservative hypersurfaces (the case \ell=0) to the higher-orderL_\ell-framework in Lorentzian space forms.