Closed-shell atoms and molecules such as Hg and TlF provide some of the best low-energy tests of hadronic CP violation beyond the standard model of particle physics, which is considered to be a necessary ingredient to explain the observed excess of matter over antimatter in our universe. CP violation is expected to be strongly enhanced in octupole-deformed nuclei such as 225 Ra. Recently, closed-shell radium-containing symmetric-top molecular ions were cooled sympathetically in a Coulomb crystal [Fan et al., Phys. Rev. Lett. 126, 023002 (2021)] and shown to be well suited for precision spectroscopy in the search for fundamental physics [Yu and Hutzler, Phys. Rev. Lett. 126, 023003 (2021)]. In closed-shell molecules hadronic CP violation contributes to a net electric dipole moment (EDM) that violates parity and time-reversal symmetries (P,T), which is the target of measurements. To interpret experiments, it is indispensable to know the electronic-structure-enhancement parameters for the various sources of P,T violation which contribute to the net P,T-odd EDM. We employ relativistic density-functional-theory calculations to determine relevant parameters for interpretation of possible EDM measurements in RaOCH₃⁺, RaSH⁺, RaCH₃⁺, RaCN⁺, and RaNC⁺ and perform accurate relativistic coupled-cluster calculations of the Schiff-moment enhancement in RaSH⁺ to gauge the quality of the density-functional-theory approach. Finally, we project to bounds on various P,T-odd parameters that could be achievable from an experiment with RaOCH₃⁺ in the near future and assess its complementarity to experiments with Hg and TlF.