The 2-SPP networks are three-level EXOR-AND-OR forms, with EXOR gates being restricted to fan-in 2. This paper presents a heuristic algorithm for the synthesis of these networks in a form that is fully testable in the stuck-at fault model (SAFM). The algorithm extends the EXPAND-IRREDUNDANT-REDUCE paradigm of ESPRESSO in heuristic mode, and it iterates local minimization and reshape of a solution until no further improvement can be achieved. This heuristic could escape from local minima using a LAST_GASP-like procedure. Moreover, the testability of 2-SPP networks under the SAFM is studied, and the notion of EXOR-irredundancy is introduced to prove that the computed 2-SPP networks are fully testable under the SAFM. Finally, this paper reports a large set of experiments showing high-quality results with affordable run times, handling also examples whose exact solutions could not be computed.