Computational micromechanics appears of the utmost importance, especially in the current context of digital twins in mechanics of materials. The objective here is to develop an efficient solver for the simulation of geometrically complex composite microstructures involving numerous inclusions connected with the matrix through various non-linear interface behaviors. To do so, we resort to IsoGeometric Analysis, which provides increased per-degree-of-freedom accuracy, and leverage the recently introduced immersed boundary-conformal method to retrieve conformal matrix/inclusion interfaces through the construction of conformal layers from it. Then, the approach is enhanced with the Large Time INcremental method that allows to separate the non-linear interface equations from those related to the subdomains, the latter being all linear and subdomain-wise independent. It results in an immersed hybrid mixed higher-order numerical scheme that is naturally parallelizable between the different subdomains and that is flexible to treat any non-linear interface behavior. The stabilization of the formulation occurs within the bulk equations where Nitsche couplings are performed. The accuracy and efficiency of the developed algorithm are demonstrated by solving a range of non-linear examples in 2D, including different numbers of inclusions in unilateral contact, frictional contact, and delamination with the matrix of the composite microstructure.