A study of Bc+→J/ψDs+ and Bc+→J/ψDs∗+ decays using 139 fb−1 of integrated luminosity collected with the ATLAS detector from s = 13 TeV pp collisions at the LHC is presented. The ratios of the branching fractions of the two decays to the branching fraction of the Bc+→ J/ψπ+ decay are measured: B(Bc+→J/ψDs+)/B(Bc+→J/ψπ+) = 2.76 ± 0.47 and B(Bc+→J/ψDs∗+)/B(Bc+→J/ψπ+) = 5.33 ± 0.96. The ratio of the branching fractions of the two decays is found to be B(Bc+→J/ψDs∗+)/B(Bc+→J/ψDs∗+) = 1.93 ± 0.26. For the Bc+→J/ψDs∗+ decay, the transverse polarization fraction, Γ±±/Γ, is measured to be 0.70 ± 0.11. The reported uncertainties include both the statistical and systematic components added in quadrature. The precision of the measurements exceeds that in all previous studies of these decays. These results supersede those obtained in the earlier ATLAS study of the same decays with s = 7 and 8 TeV pp collision data. A comparison with available theoretical predictions for the measured quantities is presented. [Figure not available: see fulltext.].