This paper investigates the long-time behavior of an explicit fully discrete scheme for the stochastic Burgers–Huxley equation driven by an additive noise, employing the spectral Galerkin spatial discretization with a temporally tamed accelerated exponential Euler method. Through a crucial monotone condition that effectively controls the Burgers-type nonlinearity, we establish the moment bound for numerical solution, which exhibits sublinear dependence with respect to the time horizon. We apply this result to present the strong convergence rate of proposed scheme and the approximation error of invariant measure. Notably, our scheme demonstrates the computational efficiency when applied to numerically approximate the invariant measure of the original equation, despite the time dependence of numerical solution. This paper constitutes the first study of invariant measure approximation for a class of stochastic partial differential equations involving Burgers-type nonlinearity.