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We report neutron scattering measurements on Er₂Pt₂O₇, a new addition to the XY family of frustrated pyrochlore magnets. Symmetry analysis of our elastic scattering data shows that Er₂Pt₂O₇ orders into the k = 0, Γ₇ magnetic structure (the Palmer-Chalker state), at 0.38 K. This contrasts with its sister XY pyrochlore antiferromagnets Er₂Ti₂O₇ and Er₂Ge₂O₇, both of which order into Γ₅ magnetic structures at much higher temperatures, 1.2 and 1.4 K, respectively. Below T_N = 0.38 K, Er₂Pt₂O₇ displays a gapped spin-wave spectrum with an intense, flat band of excitations at lower energy and a weak, diffusive band of excitations at higher energy. The flat band is well described by classical spin-wave calculations, but these calculations also predict sharp dispersive branches at higher energy, a striking discrepancy with the experimental data. This, in concert with the strong suppression of T_N, is attributable to enhanced quantum fluctuations due to phase competition between the Γ₇ and Γ₅ states that border each other within a classically predicted phase diagram.