Relativistic magnetic reconnection is a powerful agent through which magnetic energy can be tapped in astrophysics, energizing particles that then produce observed radiation. In some systems, the highest energy photons come from particles Comptonizing an ambient radiation bath supplied by an external source. If the emitting particle energies are high enough, this inverse Compton (IC) scattering enters the Klein-Nishina regime, which differs from the low-energy Thomson IC limit in two significant ways. First, radiative losses become inherently discrete, with particles delivering an order-unity fraction of their energies to single photons. Second, Comptonized photons may pair-produce with the ambient radiation, opening up another channel for radiative feedback on magnetic reconnection. We analytically study externally illuminated highly magnetized reconnecting systems for which both of these effects are important. We identify a universal (initial magnetization-independent) quasi-steady state in which gamma-rays emitted from the reconnection layer are absorbed in the upstream region, and the resulting hot pairs dominate the energy density of the inflow plasma. However, a true pair cascade is unlikely, and the number density of created pairs remains subdominant to that of the original plasma for a wide parameter range. Future particle-in-cell simulation studies may test various aspects. Pair-regulated Klein-Nishina reconnection may explain steep spectra (quiescent and flaring) from flat-spectrum radio quasars and black hole accretion disc coronae.