Black hole and neutron star environments often comprise collisionless plasmas immersed in strong magnetic fields and intense baths of low-frequency radiation. In such conditions, relativistic magnetic reconnection can tap the magnetic field energy, accelerating high-energy particles that rapidly cool by inverse Compton (IC) scattering the dense photon background. At the highest particle energies reached in bright gamma-ray sources, IC scattering can stray into the Klein-Nishina regime. Here, the Comptonized photons exceed pair-production threshold with the radiation background and may thus return their energy to the reconnecting plasma as fresh electron-positron pairs. To reliably characterize observable signatures of such Klein-Nishina reconnection, in this work, we present first-principles particle-in-cell simulations of pair-plasma relativistic reconnection coupled to Klein-Nishina and pair-production physics. The simulations show substantial differences between the observable signatures of Klein-Nishina reconnection and reconnection coupled only to low-energy Thomson IC cooling (without pair production). The latter regime exhibits strong harder-when-brighter behaviour; the former involves a stable spectral shape independent of overall brightness. This spectral stability is reminiscent of flat-spectrum radio quasar (FSRQ) GeV high states, furnishing evidence that Klein-Nishina radiative physics operates in FSRQs. The simulated Klein-Nishina reconnection pair yield spans from low to order-unity and follows an exponential scaling law in a single governing parameter. Pushing this parameter beyond its range studied here might give way to a copious pair-creation regime. Besides FSRQs, we discuss potential applications to accreting black hole X-ray binaries, the M87$^*$ magnetosphere, and gamma-ray binaries.