We focus on a family on noncommutative nonassociative substructural logics enriched with subexponentials, unary modalities licensing local application of structural rules. Primarily, we prove upper and lower bounds on complexity of provability in a broad class of these logics. To uniformly frame these results, we present two sets of rules, one intuitionistic and one classical, with the intention of capturing a broad subset of existing logics as fragments. We also prove equivalences of these systems in natural settings.