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Oxygen-isotope effects have been studied in half-doped A-site disordered Ba_0.5Ln_0.5MnO₃ with different ionic radii of the A-site (Ba_0.5Ln_0.5). We find that the ground states of the half-doped manganites strongly depend on the A-site ionic radii. Below a critical ionic radius, the ground state is spin-glass like and the freezing temperature T_SG is independent of the oxygen-isotope mass. Above the critical ionic radius, the ground state is ferromagnetic and the Curie temperature T_C depends strongly on the oxygen-isotope mass. As the ionic radius increases, the Curie-temperature increases while the oxygen-isotope exponent decreases monotonically. The current results are difficult to understand in terms of the double-exchange model along with strong lattice polaronic effects.