Topology under non-Hermitian dynamics is encoded in the phase of the bulk evolution, yet strong dissipation suppresses the amplitudes carrying it. We resolve this using a programmable photonic integrated circuit that implements such dynamics in synthetic momentum space through unitary dilation, with the phase recovered by phase-shifted interferometry. For the non-Hermitian Su-Schrieffer-Heeger model, the protocol distinguishes trivial and non-trivial Zak phases and yields a coherence winding q = ±1 induced by an exceptional point. Extending to a synthetic torus via a Rice-Mele pump gives the first Chern number Ch1 = 0, 1 for trivial and non-trivial cycles, showing a programmable route to momentum-space topology under strongly dissipative non-Hermitian dynamics.