Operational advantage without CHSH violation in qubit illumination

We study an exactly solvable qubit model of entanglement-assisted target detection in which a generalized Bell probe interrogates a lossy target while the retained idler decoheres by dephasing, of overlap gamma, and relaxation, of strength eta. The figure of merit is the Helstrom distinguishability, compared against an optimal separable-qubit benchmark at fixed mean signal excitation that we prove bounds every separable signal-idler qubit probe of that energy. Under dephasing alone every gamma > 0 violates CHSH while a wide range of those states confers no advantage, so a direct CHSH violation is not sufficient. Relaxation supplies the converse: applying the Horodecki criterion in both of its branches, the protocol can outperform every separable probe although the shared state violates no CHSH inequality, and in part of the parameter space -- in both branches -- no gamma violates it at all. This concerns the direct two-setting test only: we prove that every entangled member of the family possesses hidden CHSH nonlocality under stochastic local filtering, by an explicit filter family whose filtered CHSH value has supremum 2*sqrt(1 + gamma^2), so advantage without a direct violation coexists with universal hidden nonlocality. We then prove that EPR steering in the signal-to-idler direction is necessary across the entire advantage domain -- the idler can always be steered to a pure state -- while a three-setting witness certifies it on about 72% of the domain at kappa = 1 using only three measurement settings. For a thermal target we prove gamma_crit^2 >= 1/4 throughout the passive range N_B in [0, 1/2], with the degenerate zero-threshold branch absent there. All analytic results are accompanied by an independent numerical verification archived with this record.