We demonstrate that electron spin relaxation in GaAs in the proximity of a Fe/MgO layer is dominated by interaction with an exchange-driven hyperfine field at temperatures below 60 K. Temperature-dependent spin-resolved optical pump-probe spectroscopy reveals a strong correlation of the electron spin relaxation with carrier freeze-out, in quantitative agreement with a theoretical interpretation that at low temperatures the free-carrier spin lifetime is dominated by inhomogeneity in the local hyperfine field due to carrier localization. As the regime of large nuclear inhomogeneity is accessible in these heterostructures for magnetic fields <3 kG, inferences from this result resolve a long-standing and contentious dispute concerning the origin of spin relaxation in GaAs at low temperature when a magnetic field is present. Further, this improved fundamental understanding clarifies the importance of future experiments probing the time-dependent exchange interaction at a ferromagnet-semiconductor interface and its consequences for spin dissipation and transport during spin pumping.