While two-photon exchange (TPE) effects in space-like nucleon electromagnetic form factors have been extensively studied for over two decades, their manifestations in the time-like region have yet to be definitively established. We propose the isospin-violating decay J/{}^0 and its charge-conjugate process as a sensitive probe of TPE contributions to time-like hyperon electromagnetic form factors. This channel provides a clean laboratory for isolating TPE effects through sensitive and robust polarization and spin-correlation observables, which are particularly enhanced by the -^0 mass splitting. Combining a dispersion-relation analysis of the one-photon-exchange contribution with explicit loop calculations of the TPE amplitudes, we perform a quantitative study of TPE effects in the threshold region. Our results demonstrate that the considered observables can provide unambiguous signatures of TPE contributions and present predictions for their size together with the associated theoretical uncertainties.