This work investigates the selection principle of the critical speed in a three-species Lotka–Volterra competitive system with nonlocal diffusion. The system incorporates symmetric convolution kernels for interspecific couplings and, after a suitable change of variables, is recast into a cooperative framework. By applying the method of super- and subsolutions, we construct a novel pair of upper and lower solutions for the traveling-wave ordinary differential equations. This construction furnishes new analytic criteria that distinguish between linear and nonlinear selection of the minimal propagation speed. In particular, our results show that under suitable parameter regimes, the minimal wave speed coincides the linear prediction from the system's unstable equilibrium (linear determinacy), whereas sufficiently strong competition among the three species causes the minimal wave to travel faster than the linear theory predicts (nonlinear determinacy). These findings advance the understanding of wave speed selection in reaction–diffusion systems with nonlocal interactions by delineating the boundary between linear (pulled) and nonlinear (pushed) invasion fronts.