The magnetically frustrated manganese nitride antiperovskite family displays significant changes of entropy under changes in hydrostatic pressure near a first-order antiferromagnetic to paramagnetic phase transition that can be useful for the emerging field of solid-state barocaloric cooling. In previous studies, the transition hysteresis has significantly reduced the reversible barocaloric effects (BCE). Here we show that the transition hysteresis can be tailored through quaternary alloying in the Mn3Cu$_{1-x}$Sn$_{x}$N system. We find the magnitude of hysteresis is minimised when Cu and Sn are equiatomic (x = 0.5) reaching values far less than previously found for Mn3AN ($A = $ Pd, Ni, Ga, Zn), whilst retaining entropy changes of the same order of magnitude. These results demonstrate that reversible BCE are achievable for p < 100 MPa in the Mn3(A, B)N family and suggest routes to modify the transition properties in compounds of the same family.