The validity of a variational principle for nonequilibrium steady states proposed by Evans and Baranyai [Phys. Rev. Lett. 67, 2597 (1991)] is investigated in the case of a dilute binary mixture described by the well-known Groos-Krook kinetic model. We construct a perturbation solution around the unconstrained shear flow state and evaluate the phase-space compression factor, the temperature ratios, and the nonlinear shear viscosity up to the first-order approximation. All these quantities are nonlinear functions of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). It is shown that this principle does not hold exactly, although deviations from it are small in some situations for not very large shear rates. The calculations presented here extend previous results derived for a single dilute gas.