A new O(N) method for the iterative treatment of connected triple substitutions in the framework of local coupled cluster theory is introduced here, which is the local equivalent of the canonical CCSDT-1b method. The effect of the triple substitutions is treated in a self-consistent manner in each coupled cluster iteration. As for the local (T) method presented earlier in this series the computational cost of the method scales asymptotically linear with molecular size. The additional computational burden due to the involvement of triples in each coupled cluster iteration hence is not nearly as dramatic as for the parental canonical method, where it implies an increase in the computational complexity of the coupled cluster iteration from O(N-6) to O(N-7). The method has certain advantages in comparison to the perturbative a posteriori treatment of connected triples (T) for cases where static correlation effects start to play a role. It is demonstrated that molecules with about 100 atoms and 1000 basis functions can be treated with the local CCSDT-1b method, i.e., at a level beyond local CCSD(T). The new local coupled cluster methods introduced here and in previous papers of this series are applied in a study on the energetics of the Bergman auto-cyclization and retro-Bergman ring opening of an azaenediyne derivate, which was recently proposed as a promising candidate for anti-cancer drug development.