Inverse problems are central to material design. While numerous studies have focused on designing microstructures by inverting structure-property linkages for various material systems, such efforts stop short of providing realizable paths to manufacture such structures. Accomplishing the dual task of designing a microstructure and a feasible manufacturing pathway to achieve a target property requires inverting the complete process-structure-property linkage. However, this inversion is complicated by a variety of challenges such as inherent microstructure stochasticity, high-dimensionality, and ill-conditioning of the inversion. In this work, we propose a Bayesian framework leveraging a lightweight flow-based generative approach for the stochastic inversion of the complete process-structure-property linkage. This inversion identifies a solution distribution in the processing parameter space; utilizing these processing conditions realizes materials with the target property sets. Our modular framework readily incorporates the output of stochastic forward models as conditioning variables for a flow-based generative model, thereby learning the complete joint distribution over processing parameters and properties. We demonstrate its application to the multi-objective task of designing processing routes of heterogeneous materials given target sets of bulk elastic moduli and thermal conductivities.