I’m not sure what you’re asking here since it seems like you’ve answered your own question in your question.
Unless I’ve made a stupid mistake: If \sigma_h = \nabla \vec{u}_h - p_h \mathrm{I}, with your Taylor-Hood element, \vec{u}_h \in [V^{p}_{CG}]^{d}, \nabla \vec{u}_h \in [V^{p-1}_{DG}]^{d\times d}, and p_h \in V^{p-1}_{CG}. So, their linear combination is housed in \sigma_h \in [V^{p-1}_{DG}]^{d\times d}.
Things will get trickier if you incorporate a viscosity function.