We present a cost-effective treatment of the triple excitation amplitudes in the time-dependent optimized coupled-cluster (TD-OCC) framework called TD-OCCDT(4) for studying intense laser-driven multielectron dynamics. It considers triple excitation amplitudes correct up to the fourth-order in many-body perturbation theory and achieves a computational scaling of O(N7), with N being the number of active orbital functions. This method is applied to the electron dynamics in Ne and Ar atoms exposed to an intense near-infrared laser pulse with various intensities. We benchmark our results against the TD complete-active-space self-consistent field (TD-CASSCF), TD-OCC with double and triple excitations (TD-OCCDT), TD-OCC with double excitations (TD-OCCD), and TD Hartree-Fock (TDHF) methods to understand how this approximate scheme performs in describing nonperturbatively nonlinear phenomena, such as field-induced ionization and high-harmonic generation. We find that the TD-OCCDT(4) method performs equally well as the TD-OCCDT method, almost perfectly reproducing the results of the fully correlated TD-CASSCF with a more favorable computational scaling.