Non-Hermitian quantum many-body physics has emerged as a frontier at the intersection of condensed matter and ultracold atom research, with cold atomic gases providing a highly tunable platform for probing exotic non-Hermitian many-body effects. Motivated by these advances, we investigate a two-dimensional s-wave interacting Fermi gas subjected to both spin-orbit coupling (SOC) and engineered single-particle dissipation. Employing a non-Hermitian mean-field framework, we systematically explore the superfluid ground-state properties across two distinct pairing channels: the conventional BCS state with zero center-of-mass momentum and the finite-momentum FFLO-type state. Our results reveal a remarkable dual regulatory role of dissipation: it enhances conventional BCS superfluidity—manifested by a pronounced increase in the pairing order parameter and a concomitant reduction in the energy difference between the superfluid and normal states—while concurrently suppressing finite-momentum superfluid ordering. Mechanistically, the enhancement is attributed to a reduced single-particle energy gap and a dissipation-induced increase in the Fermi-level density of states, both of which facilitate more efficient Cooper pairing. In contrast, the suppression of finite-momentum superfluidity originates from the dissipation-driven destruction of Fermi surface asymmetry, which is initially established by the real Zeeman field; this disruption effectively eliminates the momentum-space imbalance essential for finite-momentum pairing, steering the system toward a symmetric BCS ground state. Our findings establish non-Hermitian dissipation as a powerful tuning knob for superfluid manipulation and offer experimentally testable predictions for exploring emergent physics in open quantum many-body systems.