We investigate the interplay of two-body and three-body recombination in a hybrid atom-ion system consisting of a single trapped ion immersed in a two-component Fermi gas of atoms near an atom-ion Feshbach resonance. By tuning the internal state composition of the Fermi gas at constant density and temperature, we isolate the role of quantum statistics in atom-atom-ion collisions. The measured ion loss rate exhibits a pronounced nonlinear dependence on internal state composition, revealing a reduced contribution of recombination pathways involving identical fermions already at the level of experimental observables. The observations are consistent with a two-step recombination picture and an adiabatic hyperspherical approach, where antisymmetrization restricts the available entrance channels and gives rise to interference between indistinguishable recombination pathways. Our work establishes atom-ion systems as a platform for controlling three-body collisions via quantum statistics and demonstrates that exchange-symmetry effects remain robust even under thermal averaging that obscures the underlying threshold-law behavior arising from centrifugal barriers.