The two-dimensional $O(3)$ nonlinear sigma model is a well known toy model for studying non-perturbative phenomena in quantum field theory. A central challenge is the renormalization of the energy-momentum tensor, which is complicated by the nonlinear realization of the $O(3)$ symmetry leading to non-trivial operator mixing patterns, and by large discretization artifacts affecting the determination of renormalization constants. We present results for the renormalization constants in the non-singlet sector, employing a modified lattice action with shifted boundary conditions and defining the renormalized coupling through the gradient flow. While we obtain a precise determination of the relative mixing constant $z_T$, the overall normalization $Z_T$ remains inaccessible due to large discretization artifacts. We discuss the origins of these difficulties and outline possible paths forward.