Metastable states in the antiprotonic helium isotopic system p̄³He⁺ were studied. Vibrational energy levels for the total angular momentum J = 30-38 were calculated using a three-dimensional finite element method. The dipole transition probability formalism was used to calculate the radiative decay rates of experimentally relevant transitions. Relativistic and QED corrections were calculated. Isotope effects were studied based on a comparison of our presented data with recently published results for the p̄⁴He⁺ system.