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We present the discovery of the radio afterglow and near-infrared (NIR) counterpart of the Swift short GRB 200522A, located at a small projected offset of $\approx 1$ kpc from the center of a young, star-forming host galaxy at $z=0.5536$. The radio and X-ray luminosities of the afterglow are consistent with those of on-axis cosmological short GRBs. The NIR counterpart, revealed by our HST observations at a rest-frame time of $\approx2.3$ days, has a luminosity of $\approx (1.3-1.7) \times 10^{42}$ erg s$^{-1}$. This is substantially lower than on-axis short GRB afterglow detections, but is a factor of $\approx 8$-$17$ more luminous than the kilonova of GW170817, and significantly more luminous than any kilonova candidate for which comparable observations exist. The combination of the counterpart's color ($i-y = -0.08\pm 0.21$; rest-frame) and luminosity cannot be explained by standard radioactive heating alone. We present two scenarios to interpret the broad-band behavior of GRB 200522A: a synchrotron forward shock with a luminous kilonova (potentially boosted by magnetar energy deposition), or forward and reverse shocks from a $\approx14^{\circ}$, relativistic ($Γ_0 \gtrsim 80$) jet. Models which include a combination of enhanced radioactive heating rates, low-lanthanide mass fractions, or additional sources of heating from late-time central engine activity may provide viable alternate explanations. If a stable magnetar was indeed produced in GRB 200522A, we predict that late-time radio emission will be detectable starting $\approx 0.3$-$6$ years after the burst for a deposited energy of $\approx 10^{53}$ erg. Counterparts of similar luminosity to GRB 200522A associated with gravitational wave events will be detectable with current optical searches to $\approx\!250$ Mpc.