The construction of theta-local superfield Lagrangian BRST quantization in non-Abelian hypergauges for generic gauge theories based on the action principle is examined in the case of reducible local superfield models (LSM) on the basis of embedding a gauge theory into a special theta-local superfield model with antisymplectic constraints and a Grassmann-odd time parameter theta. We examine the problem of establishing a new correspondence between the odd-Lagrangian and odd-Hamiltonian formulations of a local LSM in the case of degeneracy of the Lagrangian description with respect to derivatives over theta of generalized classical superfields A I(theta). We also reveal the role of the nilpotent BRST-BFV charge for a formal dynamical system corresponding to the BV-BFV dual description of an LSM.