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Active Galactic Nuclei in general, and the supermassive black hole in M87 in particular, show bright and rapid gamma-ray flares up to energies of 100 GeV and above. For M87, the flares show multiwavelength components, and the variability timescale is comparable to the dynamical time of the event horizon, suggesting that the emission may come from a compact region nearby the nucleus. However, the emission mechanism for these flares is not well understood. Recent high-resolution general-relativistic magnetohydrodynamic simulations show the occurrence of episodic magnetic reconnection events that can power flares nearby the black hole event horizon. In this work, we analyze the radiative properties of the reconnecting current layer under the extreme plasma conditions applicable to the black hole in M87 from the first principles. We show that abundant pair production is expected in the vicinity of the reconnection layer, to the extent that the produced secondary pair plasma dominates the reconnection dynamics. Using analytic estimates backed by two-dimensional particle-in-cell simulations we demonstrate that in the presence of strong synchrotron cooling, reconnection can produce a hard power-law distribution of pair plasma imprinted in the outgoing synchrotron (up to a few tens of MeV) and the inverse-Compton signal (up to TeV). We produce synthetic radiation spectra from our simulations, which can be directly compared with the results of future multiwavelength observations of M87* flares.