The mass sensitivity of the vibration-rotation-inversion transitions of H_3_^16^O^+^, H_3_^18^O^+^, and D_3_^16^O^+^ is investigated variationally using the nuclear motion program trove (Yurchenko, Thiel & Jensen 2007JMoSp.245..126Y). The calculations utilize new high-level ab initio potential energy and dipole moment surfaces. Along with the mass dependence, frequency data and Einstein A coefficients are computed for all transitions probed. Particular attention is paid to the {Delta}|k|=3 and {Delta}|k-l|=3 transitions comprising the accidentally coinciding |J, K=0, v_2_=0^+^> and |J, K=3, v_2_=0^-^> rotation-inversion energy levels. The newly computed probes exhibit sensitivities comparable to their ammonia and methanol counterparts, thus demonstrating their potential for testing the cosmological stability of the proton-to-electron mass ratio. The theoretical trove results are in close agreement with sensitivities obtained using the non-rigid and rigid inverter approximate models, confirming that the ab initio theory used in the present study is adequate.